- 18 Jul '11 00:41Isn't it always better to flip more than H coins, to maximize the odds of getting exactly H heads?

Say a casino gives a bettor a chance to bet a small amount to win a bigger prize. He will get to flip a number of coins and if exactly H coins come up heads, he wins. He has to choose in advance how many coins (N) to flip, and then he has to flip all N of them (but as soon as he gets to H+1 heads, the gamble is over and he loses.)

Let H = 5 in this example. Obviously he'll choose to flip at least 5, but is 5 also the most he should choose?

Without doing any math, what N would you choose? Do you think people will tend to underestimate N, and thus tilt the odds away from what the casino knows is best? What does the math say N should be, for H = 5?

Once upon a time I knew the formulas to use for this sort of thing. - 18 Jul '11 01:00[If I may just say something off-topic: I was in a meeting the other day with 3 people of varying ages whom I had not met before and who knew each other only as work colleagues. Not sure exactly how it came up but it turned out that our four birth days were 24th June, 25th June, 26th June and 27th June, all in different years. Carry on, as you were.]
- 18 Jul '11 01:03

If I understand your question, i would want to flip the coin as few times as possible and here is why:*Originally posted by JS357***Isn't it always better to flip more than H coins, to maximize the odds of getting exactly H heads?**

Say a casino gives a bettor a chance to bet a small amount to win a bigger prize. He will get to flip a number of coins and if exactly H coins come up heads, he wins. He has to choose in advance how many coins (N) to flip, and then he has to flip all N of the ...[text shortened]... N should be, for H = 5?

Once upon a time I knew the formulas to use for this sort of thing.

If you flip a coin zero times the odds of getting zero head are 1. That's the most so that's what i would pick. But I don't think you are looking for that answer so if you flip a coin once. Half of the time you get 1 head, half of the time you get 0 heads. As you flip more the distribution spreads out. If you flip a coin twice. 1/4 of the time you get no heads 1/2 you get 1, 1/4 you get 2. If you flip 3 times. 1/8 you get no heads 3/8 you get 1 head, 3/8 you get 2 heads 1/8 you get three heads. - 18 Jul '11 02:03 / 1 edit

If you get to choose the number of coins to be tossed and the objective is to get 5 heads, then toss 9 or 10 coins (as ATY says). Probability of 5 heads: ~0.246 (i.e. 24.6% chance).*Originally posted by JS357***Isn't it always better to flip more than H coins, to maximize the odds of getting exactly H heads?**

Say a casino gives a bettor a chance to bet a small amount to win a bigger prize. He will get to flip a number of coins and if exactly H coins come up heads, he wins. He has to choose in advance how many coins (N) to flip, and then he has to flip all N of the N should be, for H = 5?

Once upon a time I knew the formulas to use for this sort of thing.

It's a binomial distribution type thingy.

If you only toss 5 coins you need 5 heads out of 5 tosses, which is far less likely. Probability is (1/2)^5 = 1/32 = 0.03125 (i.e. 3.125% chance) - 20 Jul '11 10:15

This is something like the poser that asks how many people in a room are needed before there's a better than 50/50 chance that two have the same birthday. And no, it is nowhere near 365.*Originally posted by FMF***[If I may just say something off-topic: I was in a meeting the other day with 3 people of varying ages whom I had not met before and who knew each other only as work colleagues. Not sure exactly how it came up but it turned out that our four birth days were 24th June, 25th June, 26th June and 27th June, all in different years. Carry on, as you were.]** - 20 Jul '11 23:29

I think it's only twenty-something individuals, though I don't remember for sure.*Originally posted by Wajoma***This is something like the poser that asks how many people in a room are needed before there's a better than 50/50 chance that two have the same birthday. And no, it is nowhere near 365.** - 20 Jul '11 23:38

Sorry, posted to wrong forum.*Originally posted by JS357***Isn't it always better to flip more than H coins, to maximize the odds of getting exactly H heads?**

Say a casino gives a bettor a chance to bet a small amount to win a bigger prize. He will get to flip a number of coins and if exactly H coins come up heads, he wins. He has to choose in advance how many coins (N) to flip, and then he has to flip all N of the ...[text shortened]... N should be, for H = 5?

Once upon a time I knew the formulas to use for this sort of thing. - 27 Jul '11 17:32

What year/denomination is the coin?*Originally posted by JS357***Isn't it always better to flip more than H coins, to maximize the odds of getting exactly H heads?**

Say a casino gives a bettor a chance to bet a small amount to win a bigger prize. He will get to flip a number of coins and if exactly H coins come up heads, he wins. He has to choose in advance how many coins (N) to flip, and then he has to flip all N of the ...[text shortened]... N should be, for H = 5?

Once upon a time I knew the formulas to use for this sort of thing.