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35 members have voted

GM Joined: Jun 16, 2021
• Posts: 37
January 26th, 2022 at 8:23:37 AM permalink
Quote: Ace2

Henry and Tom decide to bet on a coin flip. Henry wins on heads, Tom wins on tails.

It’s \$1 per flip and they are really bored, so they decide to do one million flips. At the end of the session, the loser will write a check to the winner for the final balance.

What is the expected value and standard deviation of the check amount ?

If we use the normal approximation to the binomial distribution, we get that the expected value of the check is about sqrt(2n/π)≈0.7978845*sqrt(n) and the standard deviation is about sqrt((π-2)n/π)≈0.6028103*sqrt(n). For n=1000000, this gives expected value \$797.8845 and standard deviation \$602.8103. The values I calculated without approximation are \$797.8843 and \$602.8105.

A related question (it is easy if you look at it at the right way): show that the expected value of the check is the same for n flips and for n-1 flips, if n is even.
Ace2 Joined: Oct 2, 2017
• Posts: 1241
January 26th, 2022 at 8:44:57 AM permalink
Thanks GM, that is the correct answer for the check amount. Still reviewing your SD answer, which is very close to mine

The average check amount is (1,000,000 * 2 / π)^.5 = \$797.88

I described this method back in April (link below). At that time the Wizard’s reply was that I was “overthinking this”, but since then he published a pdf and Ask the Wizard problem using this method

The average settlement amount is a very useful number. For instance, let’s say you’re going to Vegas for the weekend and expect to play 600 blackjack hands at \$100 per hand. Your expectation is to lose \$300 +/- 2,800, but that -\$300 is basically meaningless…it’s a very long term average of results. But take \$2,800 * (2/π)^.5 to get an average settlement amount of about \$2,200 and that is much more useful information…it tells you that your result for the weekend will probably resemble something like -\$2,500 or +\$1,900. As mentioned, I originally posted about the average settlement amount earlier to get a practical figure like this
It’s all about making that GTA
ThatDonGuy Joined: Jun 22, 2011
• Posts: 5165
January 26th, 2022 at 10:51:18 AM permalink
Quote: GM

Quote: Ace2

Henry and Tom decide to bet on a coin flip. Henry wins on heads, Tom wins on tails.

It’s \$1 per flip and they are really bored, so they decide to do one million flips. At the end of the session, the loser will write a check to the winner for the final balance.

What is the expected value and standard deviation of the check amount ?

If we use the normal approximation to the binomial distribution, we get that the expected value of the check is about sqrt(2n/π)≈0.7978845*sqrt(n) and the standard deviation is about sqrt((π-2)n/π)≈0.6028103*sqrt(n). For n=1000000, this gives expected value \$797.8845 and standard deviation \$602.8103. The values I calculated without approximation are \$797.8843 and \$602.8105.

A related question (it is easy if you look at it at the right way): show that the expected value of the check is the same for n flips and for n-1 flips, if n is even.

Shows you how much I know, at least when it comes to normal distributions.

Is there an online source that shows how you came up with those mean and SD values?
Ace2 Joined: Oct 2, 2017
• Posts: 1241
January 27th, 2022 at 4:32:26 PM permalink
Quote: ThatDonGuy

Quote: GM

Quote: Ace2

Henry and Tom decide to bet on a coin flip. Henry wins on heads, Tom wins on tails.

It’s \$1 per flip and they are really bored, so they decide to do one million flips. At the end of the session, the loser will write a check to the winner for the final balance.

What is the expected value and standard deviation of the check amount ?

If we use the normal approximation to the binomial distribution, we get that the expected value of the check is about sqrt(2n/π)≈0.7978845*sqrt(n) and the standard deviation is about sqrt((π-2)n/π)≈0.6028103*sqrt(n). For n=1000000, this gives expected value \$797.8845 and standard deviation \$602.8103. The values I calculated without approximation are \$797.8843 and \$602.8105.

A related question (it is easy if you look at it at the right way): show that the expected value of the check is the same for n flips and for n-1 flips, if n is even.

Shows you how much I know, at least when it comes to normal distributions.

Is there an online source that shows how you came up with those mean and SD values?

The normal distribution has a height of 1/(2π)^.5 at x=0, so that’s it’s mean value. For the coin flip problem, we are only looking at one side of the distribution (since we don’t care if it’s + or -) but we are now saying that one side is the entire distribution. It still has to total 1, so the mean becomes 2 * 1/(2π)^.5 = (2/π)^.5

(2π)^.5 also appears in Stirling’s formula for factorials :
(2πn)^.5 * (n/e)^n. Probably appears in many places
Last edited by: Ace2 on Jan 27, 2022
It’s all about making that GTA
ThatDonGuy Joined: Jun 22, 2011
• Posts: 5165
January 27th, 2022 at 5:06:56 PM permalink
Quote: Ace2

Quote: ThatDonGuy

Shows you how much I know, at least when it comes to normal distributions.

Is there an online source that shows how you came up with those mean and SD values?

The normal distribution has a height of 1/(2π)^.5 at x=0, so that’s it’s mean value.

Er...huh? Isn't the mean of a normal distribution zero?

I can get sqrt(2/π) as the mean by dividing the integral over 0 to positive infinity of x f(x) dx by the integral over 0 to positive infinity of f(x) dx, where f(x) = 1 / sqrt(2π) * e^(-x%2/2), but I am looking for an easier way to get that number.

Oh, and while I'm in pedantic mode: "it's mean value" means "it is mean value." The word for "something belonging to it" - "its" - does not have an apostrophe.