Find the Probability of Winning a Coin Toss Game | GRE Math Practice

[lhuyvn]

Hi members,
I have traveled this forum sometimes, But this is my first question. I hope to get your help so that I can prepare better for my GRE Math test.

Following is my question.

In a game two players take turns tossing a fair coin; the winner is the firt one to toss a head. The probability that the player who makes the first toss wins the game is:
A)1/4
B)1/3
C)1/2
D)2/3
E)3/4

Thanks in advance.
LuuTruongHuy

 

[Janitor]

It has to be an advantage to make the first toss, so you can immediately rule out (A), (B), and (C).

Here are the sequences that give the win to the first player:

H
TTH
TTTTH
TTTTTTH
TTTTTTTTH
...

Just add the probabilities for the sequences above:

Probability of first player winning = (1/2) + (1/2)^3 + (1/2)^5 + ...

The infinite sum works out to 2/3, so (D) is the right answer.

 

[Gokul43201]

It has to be an advantage to make the first toss, so you can immediately rule out (A), (B), and (C).

Here are the sequences that give the win to the first player:

H
TTH
TTTTH
TTTTTTH
TTTTTTTTH
...

Just add the probabilities for the sequences above:

Probability of first player winning = (1/2) + (1/2)^3 + (1/2)^5 + ...

The infinite sum works out to 2/3, so (D) is the right answer.

Also, this sum is clearly smaller than (1/2) + [(1/2)^3 +(1/2)^4 + (1/2)^5 +...]

and the number inside [ ] is clearly (1/2)^2 or 1/4.

So the answer would have to be less that 3/4.

This variation to the approach doesn't save a whole lot of time - maybe half a minute at most - and it may leave you feeling uncertain about picking a choice without seeing any direct evidence for its correctness.

Anyways, just thought I'd drop it in as a useful elimination trick.

 

[maverick280857]

Hi

Apparently, the same question was posted in the general math forum...so we replied to it about an hour back. (I've changed the equations a bit though they are essentially the same things).