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Comments on How do I compare lotteries' chances of winning jackpot, when they differ in the maximum number of plays?

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How do I compare lotteries' chances of winning jackpot, when they differ in the maximum number of plays?

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However, there is one way to boost your chances of winning the lottery, says [Dr. Mark] Glickman: Your odds do improve by buying more tickets for each game.

  1. Do the odds below factor in each lottery's different maximum (number of) plays? Or are these odds calculated for merely 1 play?

  2. Does each lottery's different maximum (number of) plays affect the probability of winning the jackpot?

  3. How can I deduce which lottery's jackpot is easiest to win, when they differ in the maximum plays?

  4. For example, which has the highest chance of winning : 10 plays of Lotto 649 vs. 5 plays of DAILY GRAND (assume I pick $7 million lump sum) vs. 2 plays of DAILY KENO?

Data for 3 lotteries from the Ontario Lottery and Gaming Corporation

LOTTO 649 DAILY GRAND DAILY KENO
Jackpot $5 million $1,000/day for life or \$7 million lump sum $2.5 million
Odds of winning jackpot $\dfrac1{13,983,816}$ $\dfrac1{13,348,188}$ $\dfrac1{2,147,181}$
Matching numbers required to win jackpot 6/6 5/5 + Grand Number 10/20 on a $10 bet
Number pool 49 49 + Grand Number (7) 70
Number of tickets one person can buy 10 5 2
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4 comment threads

x-post https://www.reddit.com/r/probabilitytheory/comments/1257tqx/how_do_i_compare_lotteries_chances... (1 comment)
x-post https://www.reddit.com/r/cheatatmathhomework/comments/126g0d3/how_do_i_compare_lotteries_chanc... (1 comment)
x-post https://www.reddit.com/r/AskStatistics/comments/125bu54/how_do_i_compare_lotteries_chances_of_... (1 comment)
x-post https://www.reddit.com/r/Probability/comments/125ba7h/how_do_i_compare_lotteries_chances_of_wi... (1 comment)
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Do the odds below factor in each lottery's different maximum (number of) plays? Or are these odds calculated for merely 1 play?

You'd have to ask each lottery, but usually such published odds are the chance of any one ticket winning. That appears to be the case here too.

Does each lottery's different maximum (number of) plays affect its odds?

No. The chance of any one ticket winning are the same regardless of how many tickets you are allowed to buy. Buying more tickets increases your chance of winning, but not that of any one ticket.

How can I deduce which lottery's jackpot is easiest to win, when they differ in the maximum plays?

By comparing the chance to win per ticket. Since there is no rule that everyone always must buy the maximum number of tickets, you can only compare the probability of any one ticket winning. If you buy multiple tickets, then your overall chance of winning increases.

This is the same for any group of tickets. If all 10 people in the neighborhood each buy one ticket, then the neighborhood has the same chance of winning as an individual that bought 10 tickets.

For small numbers of tickets and small chances of winning per ticket (like all the lotteries you quote), then the chance of winning increases linearly with the number of tickets to a good approximation.

Note that the chance of winning isn't really linear with the number of tickets. Let's say each ticket has a 50% chance of winning, like flipping a coin. Two tickets have 75% chance of winning, and 5 tickets 96.9% chance.

For example, which has the highest chance of winning : 10 plays of Lotto 649 (5 million) vs. 5 plays of DAILY GRAND (assume I pick 7 million lump sum) vs. 2 plays of DAILY KENO (2.5 million)?

You can look this up yourself. Note that the payout is irrelevant to your question, which simplifies it considerably. The chance of winning has nothing to do how much you get if you happen to win.

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1 comment thread

Linearity (5 comments)
Linearity
Peter Taylor‭ wrote about 1 year ago

I would say that in the context of the question the chance of winning is linear with the number of tickets because we can assume that the tickets will not have the same numbers, and a perfect match with the selected numbers is required to win, so they cover different outcomes of a single event.

Olin Lathrop‭ wrote about 1 year ago

Peter Taylor‭ I agree that what you're saying is effectively the case, but only because the linear approximation is quite good for small numbers of tickets relative to the odds of winning. If each ticket is guaranteed to have a unique number, then it is truly linear. But then only a fixed number can be sold. If the number on each ticket is randomly generated, then the chance of winning goes up less than linearly with large numbers of tickets. For example, suppose each drawing a 6-digit decimal number is chosen as the winner. Each ticket has a 1/1,000,000 chance of winning, but you aren't guaranteed a win in 1,000,000 randomly generated tickets. Now if you can control the number of each ticket you buy then the chance of winning is linear with the number of tickets if you make sure that the number on each ticket is unique. Again though, for OP's case, the chance of winning is effectively linear with the number of tickets since the chance of duplication is tiny.

Chgg Clou‭ wrote about 1 year ago · edited 12 months ago

Thanks. I reworded my question 2. I didn't know I asked a question different from the one in my mind. I have 4 follow up questions please. 1. Please elaborate the following sentences? Please prove them mathematically? "For small numbers of tickets and small chances of winning per ticket (like all the lotteries you quote), then the chance of winning increases linearly with the number of tickets to a good approximation. Note that the chance of winning isn't really linear with the number of tickets." 2. The previous sentences feel wrong? Presuppose 2,147,181 different people buy 1 ticket each of DAILY KENO. Then by your logic, someone must win DAILY KENO, which is wrong. Just because $p$ people buy a lottery with $1/p$ odds of winning jackpot, doesn't mean that someone will win jackpot.

Chgg Clou‭ wrote about 1 year ago · edited about 1 year ago

3. "Two tickets have 75% chance of winning, and 5 tickets 96.9% chance." How did you compute these rational numbers? 4. "You can look this up yourself." I can? How? My table shows the odds for merely each play. It doesn't show odds for 10 plays of Lotto 649, 5 plays of DAILY GRAND, or 2 plays of DAILY KENO.

Peter Taylor‭ wrote almost 1 year ago

I was specifically assuming that the numbers on each ticket are not randomly generated but chosen by the player.