Comments on How to intuit P(win the same lottery twice) $= p^{2}$ vs. P(win the same lottery twice | you won the lottery once) $= p$?
Parent
How to intuit P(win the same lottery twice) $= p^{2}$ vs. P(win the same lottery twice | you won the lottery once) $= p$?
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", 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."
$\Pr(\color{crimson}{W_2} | {\color{Green}{W_1}}) = \dfrac{\color{crimson}{\Pr(W_2)} \cap \color{Green}{\Pr(W_1)}}{\color{Green}{\Pr(W_1)}} = \dfrac{\color{crimson}{\Pr(W_2)} \times \color{Green}{\Pr(W_1)}}{\color{Green}{\Pr(W_1)}} = \dfrac{\color{red}{p} \times \color{limegreen}{p}}{\color{limegreen}p} = p.$
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 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" 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).
Post
It feels contradictory for P(you win the same lottery twice) $\neq$ P(you win the same lottery twice|you won the lottery once).
Would you expect P(you win the lottery exactly zero times) = P(you win the lottery exactly zero times | you won the lottery once)?
Intuitively, why aren't these two probabilities equal?
There are at least two ways of looking at it.
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P(you win the same lottery twice|you won the lottery once) is a simpler case than most conditional probabilities. It can be straightforwardly rephrased as P(you win the lottery for a second time).
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Conditional probability always excludes some possibilities. The possibility that you never win the lottery must be taken into account when calculating P(you win the same lottery twice) but it must be ruled out when calculating P(you win the same lottery twice|you won the lottery once).
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