## 2-Coin Toss Probability Proof

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### 2-Coin Toss Probability Proof

Q: "You flip a fair coin twice. The probability of getting heads is 0.5, and tails 0.5. If E represented an event and each of the two tosses was represented by the numbers 1 and 2 respectively (i.e. H1 means getting heads on the first toss), then the possible outcomes are:

E1 = H1, H2
E2 = H1, T2
E3 = T1, H2
E4 = T1, T2

Since the coin is completely fair, we can assume that these pairs of outcomes are equally likely. That is to say, the probability of E1:

P(E1) = 1 / 4

This mistake is commonly seen in introductory Probability and Statistics Textbooks."

I talked to my professor and he said it was a really "subtle" question and not to over think it. Basically, the reasoning for why the answer is 1/4 is incomplete. That is to say there is something critical, that is missing from this argument. My guess is you cannot conclude that the "pairs of outcomes are equally likely" just because the coin is fair and each elementary outcome has a 0.5 chance of occurring. That said, even if my guess is correct I don't know what missing piece is so critical that there exists a situation where this assumption is false without it.

Sorry if I broke any rules to this forum, first post.
Risotto17
Newcomer

Posts: 2
Joined: Sun Jan 22, 2012 1:04 am

### Re: 2-Coin Toss Probability Proof

Risotto17 wrote:Q: "You flip a fair coin twice. The probability of getting heads is 0.5, and tails 0.5. If E represented an event and each of the two tosses was represented by the numbers 1 and 2 respectively (i.e. H1 means getting heads on the first toss), then the possible outcomes are:

E1 = H1, H2
E2 = H1, T2
E3 = T1, H2
E4 = T1, T2

Since the coin is completely fair, we can assume that these pairs of outcomes are equally likely. That is to say, the probability of E1:

P(E1) = 1 / 4

This mistake is commonly seen in introductory Probability and Statistics Textbooks."

I talked to my professor and he said it was a really "subtle" question and not to over think it. Basically, the reasoning for why the answer is 1/4 is incomplete. That is to say there is something critical, that is missing from this argument. My guess is you cannot conclude that the "pairs of outcomes are equally likely" just because the coin is fair and each elementary outcome has a 0.5 chance of occurring. That said, even if my guess is correct I don't know what missing piece is so critical that there exists a situation where this assumption is false without it.

Sorry if I broke any rules to this forum, first post.

The two tosses have to be independent for the probability that both be H is the product of the individual tosses be heads.

CB
CaptainBlack
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Posts: 14
Joined: Fri Jan 20, 2012 10:56 am

### Re: 2-Coin Toss Probability Proof

Interesting thought. The .50 probability of H/T not changing and being independent are indeed two different things.
So this is how liberty dies... with thunderous applause.

Erimess

Posts: 782
Joined: Sat Apr 30, 2011 6:38 am
Location: USA

### Re: 2-Coin Toss Probability Proof

CaptainBlack wrote:
Risotto17 wrote:Q: "You flip a fair coin twice. The probability of getting heads is 0.5, and tails 0.5. If E represented an event and each of the two tosses was represented by the numbers 1 and 2 respectively (i.e. H1 means getting heads on the first toss), then the possible outcomes are:

E1 = H1, H2
E2 = H1, T2
E3 = T1, H2
E4 = T1, T2

Since the coin is completely fair, we can assume that these pairs of outcomes are equally likely. That is to say, the probability of E1:

P(E1) = 1 / 4

This mistake is commonly seen in introductory Probability and Statistics Textbooks."

I talked to my professor and he said it was a really "subtle" question and not to over think it. Basically, the reasoning for why the answer is 1/4 is incomplete. That is to say there is something critical, that is missing from this argument. My guess is you cannot conclude that the "pairs of outcomes are equally likely" just because the coin is fair and each elementary outcome has a 0.5 chance of occurring. That said, even if my guess is correct I don't know what missing piece is so critical that there exists a situation where this assumption is false without it.

Sorry if I broke any rules to this forum, first post.

The two tosses have to be independent for the probability that both be H is the product of the individual tosses be heads.

CB

So the only thing missing is a proof that the two events are independent?

Would something to the effect of:
P(HnT) = P(H|T)P(T)
= P(H)P(T)
so,
P(H|T) = P(HnT) / P(T) = P(H)
and,
P(T|H) = P(TnH) / P(H) = P(T)

suffice to complete the proof?

Thanks a ton.
Risotto17
Newcomer

Posts: 2
Joined: Sun Jan 22, 2012 1:04 am

### Re: 2-Coin Toss Probability Proof

The definition of independednce is equvalent to the the reqirement that the probability of AB in that order is equal to the product of A and the probability of B.

CB
CaptainBlack
Newcomer

Posts: 14
Joined: Fri Jan 20, 2012 10:56 am