Post History
#7: Post edited
by
Chgg Clou
·
2023-04-29T06:26:43Z (over 1 year ago)
Each lottery draw is independent, with probability $0 < p ≪ 1$. $W_i$ denotes the event of winning the $i^{th}$ draw of the same lottery.- #### $\Pr(\color{green}{W_1} \cap \color{crimson}{W_2}) = p^{2}$
- ["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/), because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)
- #### $\Pr(\color{crimson}{W_2} | {\color{Green}{W_1}}) = \dfrac{\color{crimson}{\Pr(W_2)} \cap \color{Green}{\Pr(W_1)}}{\color{Green}{\Pr(W_1)}} = \dfrac{\color{crimson}{\Pr(W_2)} \times \color{Green}{\Pr(W_1)}}{\color{Green}{\Pr(W_1)}} = \dfrac{\color{red}{p} \times \color{limegreen}{p}}{\color{limegreen}p} = p.$
- >[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
>[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event (1 in 176 million).](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/)- ### My misgivings
$\Pr(\color{green}{W_1} \cap \color{crimson}{W_2})eq \Pr(\color{crimson}{W_2} | \color{green}{W_1})$ feels contradictory. Intuitively, why don’t these two probabilities equal other?- Doesn’t a player's ["odds for winning the next time are the same as if they'd never played before"](https://math.stackexchange.com/a/3209757) contradict $\Pr(\color{green}{W_1} \cap \color{crimson}{W_2})= p^{2}$. How can I reconcile these 2 probabilities intuitively?
- $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ feels deceitful. Given that you won the lottery once, $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ implies that winning again is the same probabilistically as your first win! But this is wrong! $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ _OUGHT_, but wholly fails to, disclose that winning any lottery twice is way less probable than winning it once.
- I ask not about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).
- Each lottery draw is independent, with probability $0 < p ≪ 1$. $W_i$ denotes the event of winning the $i^{th}$ draw of the same lottery. I seek intuition, not asking about computations.
- #### $\Pr(\color{green}{W_1} \cap \color{crimson}{W_2}) = p^{2}$
- ["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/), because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)
- #### $\Pr(\color{crimson}{W_2} | {\color{Green}{W_1}}) = \dfrac{\color{crimson}{\Pr(W_2)} \cap \color{Green}{\Pr(W_1)}}{\color{Green}{\Pr(W_1)}} = \dfrac{\color{crimson}{\Pr(W_2)} \times \color{Green}{\Pr(W_1)}}{\color{Green}{\Pr(W_1)}} = \dfrac{\color{red}{p} \times \color{limegreen}{p}}{\color{limegreen}p} = p.$
- >[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
- >[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/).
- ### My misgivings
- $\Pr(\color{green}{W_1} \cap \color{crimson}{W_2})
- eq \Pr(\color{crimson}{W_2} | \color{green}{W_1})$ feels contradictory. Intuitively, why doesn't $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p^2$?
- Doesn’t a player's ["odds for winning the next time are the same as if they'd never played before"](https://math.stackexchange.com/a/3209757) contradict $\Pr(\color{green}{W_1} \cap \color{crimson}{W_2})= p^{2}$. How can I reconcile these 2 probabilities intuitively?
- $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ feels deceitful. Given that you won the lottery once, $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ implies that winning again is the same probabilistically as your first win! But this is wrong! $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ _OUGHT_, but wholly fails to, disclose that winning any lottery twice is way less probable than winning it once.
- I ask not about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).
#6: Post edited
by
Chgg Clou
·
2023-04-27T07:24:58Z (over 1 year ago)
How to intuit P(win the same lottery twice) $= p^{-2}$ vs. P(win the same lottery twice | you won the lottery once) $= p^{-1}$?
- How to intuit P(win the same lottery twice) $= p^{2}$ vs. P(win the same lottery twice | you won the lottery once) $= p$?
Presume that each lottery draw is independent with probability $p$. I'm seeking intuition here. I'm not asking about computations, which I did below.#### Pr(you win the same lottery twice) =["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/) $= p^{-2}$, because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)#### Pr(you win the same lottery twice | you won the lottery once)- >[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
- >[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event (1 in 176 million).](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/)
I can prove this. $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{Green}{\text{you won lottery once}}}) = \dfrac{\Pr(\text{you win the same lottery twice $\cap$ you won lottery once})}{\color{Green}{\Pr(\text{you won lottery once})}} = \dfrac{\Pr(\text{you win the same lottery twice})}{\color{Green}{\Pr(\text{you won lottery once}})} = \dfrac{p^{-2}}{p^{-1}} = \dfrac1p$.## My misgivingsIt feels contradictory for P(you win the same lottery twice) $\neq$ P(you win the same lottery twice|you won the lottery once). Intuitively, why aren't these two probabilities equal?In other words, a player's ["odds for winning the next time are the same as if they'd never played before"](https://math.stackexchange.com/a/3209757) contradicts $\Pr($you win the same lottery twice$)= p^{-2}$. How can I reconcile these 2 probabilities intuitively?I feel that $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}})$ _OUGHT_ $= \dfrac1{p^2}$, because this Conditional Probability _OUGHT_ capture that P(winning the same lottery twice) $= p^{-2}$.At the least, $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}}) = \dfrac1p$ feels misleading. Given that you won the lottery once, this Conditional Probability implies that winning again is the same probabilistically as your first win! But this is wrong! Winning the same lottery twice is way less probable than your first win!I'm not asking about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).
- Each lottery draw is independent, with probability $0 < p ≪ 1$. $W_i$ denotes the event of winning the $i^{th}$ draw of the same lottery.
- #### $\Pr(\color{green}{W_1} \cap \color{crimson}{W_2}) = p^{2}$
- ["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/), because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)
- #### $\Pr(\color{crimson}{W_2} | {\color{Green}{W_1}}) = \dfrac{\color{crimson}{\Pr(W_2)} \cap \color{Green}{\Pr(W_1)}}{\color{Green}{\Pr(W_1)}} = \dfrac{\color{crimson}{\Pr(W_2)} \times \color{Green}{\Pr(W_1)}}{\color{Green}{\Pr(W_1)}} = \dfrac{\color{red}{p} \times \color{limegreen}{p}}{\color{limegreen}p} = p.$
- >[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
- >[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event (1 in 176 million).](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/)
- ### My misgivings
- $\Pr(\color{green}{W_1} \cap \color{crimson}{W_2}) \neq \Pr(\color{crimson}{W_2} | \color{green}{W_1})$ feels contradictory. Intuitively, why don’t these two probabilities equal other?
- Doesn’t a player's ["odds for winning the next time are the same as if they'd never played before"](https://math.stackexchange.com/a/3209757) contradict $\Pr(\color{green}{W_1} \cap \color{crimson}{W_2})= p^{2}$. How can I reconcile these 2 probabilities intuitively?
- $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ feels deceitful. Given that you won the lottery once, $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ implies that winning again is the same probabilistically as your first win! But this is wrong! $\Pr(\color{crimson}{W_2} | \color{green}{W_1}) = p$ _OUGHT_, but wholly fails to, disclose that winning any lottery twice is way less probable than winning it once.
- I ask not about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).
#5: Post edited
by
Chgg Clou
·
2023-04-25T06:49:29Z (over 1 year ago)
How to intuit Pr(win the same lottery twice) vs. Pr(win the same lottery twice|you won the lottery once)?
- How to intuit P(win the same lottery twice) $= p^{-2}$ vs. P(win the same lottery twice | you won the lottery once) $= p^{-1}$?
#4: Post edited
by
Chgg Clou
·
2023-04-25T05:23:24Z (over 1 year ago)
How to intuit P(win the same lottery twice) vs. P(win the same lottery twice|you won the lottery once)?
- How to intuit Pr(win the same lottery twice) vs. Pr(win the same lottery twice|you won the lottery once)?
#3: Post edited
by
Chgg Clou
·
2023-04-25T05:22:54Z (over 1 year ago)
- Presume that each lottery draw is independent with probability $p$. I'm seeking intuition here. I'm not asking about computations, which I did below.
- #### Pr(you win the same lottery twice) =
- ["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/) $= p^{-2}$, because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)
#### P(you win the same lottery twice|you won the lottery once)- >[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
- >[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event (1 in 176 million).](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/)
- I can prove this. $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{Green}{\text{you won lottery once}}}) = \dfrac{\Pr(\text{you win the same lottery twice $\cap$ you won lottery once})}{\color{Green}{\Pr(\text{you won lottery once})}} = \dfrac{\Pr(\text{you win the same lottery twice})}{\color{Green}{\Pr(\text{you won lottery once}})} = \dfrac{p^{-2}}{p^{-1}} = \dfrac1p$.
- ## My misgivings
- It feels contradictory for P(you win the same lottery twice) $\neq$ P(you win the same lottery twice|you won the lottery once). Intuitively, why aren't these two probabilities equal?
- In other words, a player's ["odds for winning the next time are the same as if they'd never played before"](https://math.stackexchange.com/a/3209757) contradicts $\Pr($you win the same lottery twice$)= p^{-2}$. How can I reconcile these 2 probabilities intuitively?
- I feel that $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}})$ _OUGHT_ $= \dfrac1{p^2}$, because this Conditional Probability _OUGHT_ capture that P(winning the same lottery twice) $= p^{-2}$.
At the least, $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}}) = \dfrac1p$ feels misleading. Pretend you won the lottery. Then this Conditional Probability implies that winning again is no different probabilistically from your first win! But this is wrong! Winning the same lottery twice is way less probable than winning once, which you did!- I'm not asking about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).
- Presume that each lottery draw is independent with probability $p$. I'm seeking intuition here. I'm not asking about computations, which I did below.
- #### Pr(you win the same lottery twice) =
- ["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/) $= p^{-2}$, because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)
- #### Pr(you win the same lottery twice | you won the lottery once)
- >[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
- >[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event (1 in 176 million).](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/)
- I can prove this. $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{Green}{\text{you won lottery once}}}) = \dfrac{\Pr(\text{you win the same lottery twice $\cap$ you won lottery once})}{\color{Green}{\Pr(\text{you won lottery once})}} = \dfrac{\Pr(\text{you win the same lottery twice})}{\color{Green}{\Pr(\text{you won lottery once}})} = \dfrac{p^{-2}}{p^{-1}} = \dfrac1p$.
- ## My misgivings
- It feels contradictory for P(you win the same lottery twice) $\neq$ P(you win the same lottery twice|you won the lottery once). Intuitively, why aren't these two probabilities equal?
- In other words, a player's ["odds for winning the next time are the same as if they'd never played before"](https://math.stackexchange.com/a/3209757) contradicts $\Pr($you win the same lottery twice$)= p^{-2}$. How can I reconcile these 2 probabilities intuitively?
- I feel that $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}})$ _OUGHT_ $= \dfrac1{p^2}$, because this Conditional Probability _OUGHT_ capture that P(winning the same lottery twice) $= p^{-2}$.
- At the least, $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}}) = \dfrac1p$ feels misleading. Given that you won the lottery once, this Conditional Probability implies that winning again is the same probabilistically as your first win! But this is wrong! Winning the same lottery twice is way less probable than your first win!
- I'm not asking about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).
#2: Post edited
by
Chgg Clou
·
2023-04-25T05:21:39Z (over 1 year ago)
- Presume that each lottery draw is independent with probability $p$. I'm seeking intuition here. I'm not asking about computations, which I did below.
- #### Pr(you win the same lottery twice) =
["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/) $= \dfrac1{p^2}$, because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)- #### P(you win the same lottery twice|you won the lottery once)
- >[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
- >[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event (1 in 176 million).](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/)
>[Their odds for winning the next time are the same as if they'd never played before.](https://math.stackexchange.com/a/3209757)I can prove this. $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}}) = \dfrac{\Pr(\text{you win the same lottery twice $\cap$ you won lottery once})}{\color{green}{\Pr(\text{you won lottery once})}} = \dfrac{\Pr(\text{you win the same lottery twice})}{\color{Green}{\Pr(\text{you won lottery once}})} = \dfrac{p^{-2}}{p^{-1}} = \dfrac1p$.- ## My misgivings
It feels contradictory for P(you win the same lottery twice) $eq$ P(you win the same lottery twice|you won the lottery once). Intuitively, why aren't these two probabilities equal?I feel that $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}})$ _OUGHT_ $= \dfrac1{p^2}$, because this Conditional Probability fails to capture that P(winning the same lottery twice) $= p^{-2}$.At the least, $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}}) = \dfrac1p$ feels misleading. Pretend you won the lottery. Then this Conditional Probability implies that winning again is no different probabilistically from winning the first time! But this is wrong! Winning the same lottery twice is way less probable than winning the first time!- I'm not asking about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).
- Presume that each lottery draw is independent with probability $p$. I'm seeking intuition here. I'm not asking about computations, which I did below.
- #### Pr(you win the same lottery twice) =
- ["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/) $= p^{-2}$, because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)
- #### P(you win the same lottery twice|you won the lottery once)
- >[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
- >[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event (1 in 176 million).](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/)
- I can prove this. $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{Green}{\text{you won lottery once}}}) = \dfrac{\Pr(\text{you win the same lottery twice $\cap$ you won lottery once})}{\color{Green}{\Pr(\text{you won lottery once})}} = \dfrac{\Pr(\text{you win the same lottery twice})}{\color{Green}{\Pr(\text{you won lottery once}})} = \dfrac{p^{-2}}{p^{-1}} = \dfrac1p$.
- ## My misgivings
- It feels contradictory for P(you win the same lottery twice) $
- eq$ P(you win the same lottery twice|you won the lottery once). Intuitively, why aren't these two probabilities equal?
- In other words, a player's ["odds for winning the next time are the same as if they'd never played before"](https://math.stackexchange.com/a/3209757) contradicts $\Pr($you win the same lottery twice$)= p^{-2}$. How can I reconcile these 2 probabilities intuitively?
- I feel that $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}})$ _OUGHT_ $= \dfrac1{p^2}$, because this Conditional Probability _OUGHT_ capture that P(winning the same lottery twice) $= p^{-2}$.
- At the least, $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}}) = \dfrac1p$ feels misleading. Pretend you won the lottery. Then this Conditional Probability implies that winning again is no different probabilistically from your first win! But this is wrong! Winning the same lottery twice is way less probable than winning once, which you did!
- I'm not asking about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).
#1: Initial revision
by
Chgg Clou
·
2023-04-25T05:13:31Z (over 1 year ago)
How to intuit P(win the same lottery twice) vs. P(win the same lottery twice|you won the lottery once)?
Presume that each lottery draw is independent with probability $p$. I'm seeking intuition here. I'm not asking about computations, which I did below.
#### Pr(you win the same lottery twice) =
["the probability of winning the lottery multiplied by winning the lottery the second time"](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxmy45/) $= \dfrac1{p^2}$, because ["the two events _are_ independent. In other words, winning the lottery once doesn't some how increase or decrease your chances of winning it a second time."](https://old.reddit.com/r/learnmath/comments/38si94/high_school_mathprobability_of_winning_a_lottery/crxq4eq/)
#### P(you win the same lottery twice|you won the lottery once)
>[“If someone already wins the lottery, then the chance that the person wins the lottery a second time will be exactly the same as the probability they win the lottery if they had not previously won the lottery before,” Harvard statistics professor Dr. Mark Glickman tells CNBC Make It.](https://www.cnbc.com/2019/05/31/harvard-prof-on-odds-of-winning-multiple-lotteries-like-these-people.html)
>[If you've _already_ won the lottery in week one, however, the odds of winning the next week will be unaffected by the outcome and so remain the same as for any other individual event (1 in 176 million).](https://old.reddit.com/r/askscience/comments/pni9s/lottery_lightning_strikes_twice/c3qro0g/)
>[Their odds for winning the next time are the same as if they'd never played before.](https://math.stackexchange.com/a/3209757)
I can prove this. $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}}) = \dfrac{\Pr(\text{you win the same lottery twice $\cap$ you won lottery once})}{\color{green}{\Pr(\text{you won lottery once})}} = \dfrac{\Pr(\text{you win the same lottery twice})}{\color{Green}{\Pr(\text{you won lottery once}})} = \dfrac{p^{-2}}{p^{-1}} = \dfrac1p$.
## My misgivings
It feels contradictory for P(you win the same lottery twice) $\neq$ P(you win the same lottery twice|you won the lottery once). Intuitively, why aren't these two probabilities equal?
I feel that $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}})$ _OUGHT_ $= \dfrac1{p^2}$, because this Conditional Probability fails to capture that P(winning the same lottery twice) $= p^{-2}$.
At the least, $\Pr({\color{crimson}{\text{you win the same lottery twice}}} | {\color{green}{\text{you won lottery once}}}) = \dfrac1p$ feels misleading. Pretend you won the lottery. Then this Conditional Probability implies that winning again is no different probabilistically from winning the first time! But this is wrong! Winning the same lottery twice is way less probable than winning the first time!
I'm not asking about [$\Pr($you win a lottery at least twice) $= 1 - \Pr($you never win) $- \Pr($you win the lottery once)](https://math.stackexchange.com/q/3233269).