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## Question

Two boys are bored, so they decide to play a coin game. They will alternate turns flipping a coin. The first one to land it on heads has to pay the other \$5 dollars. The 2nd boy, being a gentleman, gives the 1st boy a decision, to go first or second. Which has the better chance, going 1st or 2nd, of landing heads first and how?

## 11 answers to this question

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It is irrelevant who goes first or even how many times they each go. Every time you flip a coin there is a 50/50 chance and each coin flip is independent of any other flips.

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i agree with beccaf22 there is a 0.5 probability chance every flip althought mathematically if you flip first then 2 x 0.5 = 1 so theoretically the person that flips first should win

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Nope, beccaf22 is wrong; andreay, you're answer is right, but your reasoning is off.

There's some advanced math in this, so bear with me.

The one that goes first has a better chance of winning.

The probability of winning is the sum of the probabilities of each turn.

1st turn: 1/2

2nd turn (in this case, the 3rd overall flip): 1/23

3rd turn (in this case, the 5th overall flip): 1/25

So adding those: 1/2+1/23+1/25 and so on.

It's an infinite series that approaches 2/3 in value.

Therefore, the player who tosses first has almost 2x a better chance of winning.

Still surprised, try it for yourself.

Edited by dawgsrule93

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Yes....you got the numbers and answer right dawgsrule93.....but notice that the person for who the coin lands heads _pays_ the other \$5.

(and for future reference....use spoiler blocks so others can look at posts for clarification and not have the answer right in front of them.)

Going first is the better chance of getting the coin to land head first.... but going second is the way to win the money.

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You don't even have to make it that complicated. The question asks who is most likely to land heads first. The first boy is the answer because he goes first. He has a 1/2 probable chance before the second boy even gets a shot.

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You're right, i meant to say, the one who gets heads first wins \$5. OK, yeah, i shouldn't have done that, my bad.

Edited by dawgsrule93

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You're right, i meant to say, the one who gets heads first wins \$5. How do you do spoiler blocks? i am new to BrainDen. I joined yesterday.

When you post look at the left Quick Access boxes and the buttons above the typing area. Frame you're responses with Spoiler, Quotes, etc... Toggle the side panel if yours is hidden.

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The second person has the advantage. It's a 50% chance of getting heads on each throw, so the only factor that matters is who goes before who. The person that goes first has a greater chance of getting a head first, so the person going second has a better chance of winning \$5 than the first person

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Two boys are bored, so they decide to play a coin game. They will alternate turns flipping a coin. The first one to land it on heads has to pay the other \$5 dollars. The 2nd boy, being a gentleman, gives the 1st boy a decision, to go first or second. Which has the better chance, going 1st or 2nd, of landing heads first and how?

I believe the first boy has the advantage. He has a 50% chance of getting it, while the second boy has a 50% chance of even getting his 50% chance of landing heads.

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Two boys are bored, so they decide to play a coin game. They will alternate turns flipping a coin. The first one to land it on heads has to pay the other \$5 dollars. The 2nd boy, being a gentleman, gives the 1st boy a decision, to go first or second. Which has the better chance, going 1st or 2nd, of landing heads first and how?

The first person has a 2/3 chance of winning, the second has a 1/3 chance.

The probability the first wins on the first toss is 1/2, on the third toss is (1/2)^3, on the 5th toss (1/2)^5, etc.

Add these probabilities (geometric sequence) and you get 2/3.

Going second, probability you win on second toss (your first) is 1/4, 4th toss (1/2)^4, etc. sum=1/3

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Ok I think I get it now,

the thing is you are essentially asking for a series where the order is TH, then TTH, then TTTH etc. so probabilities are

1/2 (=.5)

1/2*1/2 (= 0.25)

1/2*1/2*1/2 (= 0.125) etc...

so 1st flipper is 0.5 + 0.125...

2nd flipper is 0.25 + ...

(which I am not gonna do the math for but ends up close to .67 and .33)

so the 1st flipper has about a 2x better chance of getting heads first and it is almost all accounted for by the .5 to .25 difference between the 1st and 2nd flips and relies on the fact that previous flips must all be tails...

Interestingly, when you toss a real coin (not a theoretical equal probability one) there are some interesting discrepancies with theory

the news story

the math

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