M15-31

Question

If John throws a coin until a series of three consecutive heads or three consecutive tails appears, what is the probability that the game will end on the fourth throw?

A.  \frac{1}{16}  
B.  \frac{1}{8}  
C.  \frac{3}{16}  
D.  \frac{1}{4}  
E.  \frac{3}{8}

Bunuel · Accepted Answer

Official Solution:

If John throws a coin until a series of three consecutive heads or three consecutive tails appears, what is the probability that the game will end on the fourth throw?

A.  \frac{1}{16}  
B.  \frac{1}{8}  
C.  \frac{3}{16}  
D.  \frac{1}{4}  
E.  \frac{3}{8}

Calculate the probability that either of the two sequences,  THHH  or  HTTT , will occur. The probability of  THHH  occurring =  (\frac{1}{2})^4 = \frac{1}{16} . The probability of  HTTT  occurring =  (\frac{1}{2})^4 = \frac{1}{16} . The probability of either  THHH  or  HTTT  occurring =  \frac{1}{16} + \frac{1}{16} = \frac{2}{16} = \frac{1}{8} .

Answer: B

mejia401 · Answer

Since there are two options, Beroulli's formula can also be applied:

\frac{2!}{1!1!}*\frac{1}{2}^{4-1}*\frac{1}{2}^{4-3}

2*\frac{1}{8}*\frac{1}{2}

\frac{2}{16}

B.

vivekgautam1 · Answer

Hi Bunuel,

Here's how i am approaching this question.

The question is asking - out of 4 throws, what's the probability of getting 3 consecutive heads or 3 consecutive tails.

Total # of arrangements = 2 * 2 * 2 * 2 = 16

Now, 3 consecutive heads can be like - THHH or HHHT
Similarly, 3 consecutive tails can be like - HTTT or TTTH.

So, total # of favourable outcomes = 2 + 2 = 4.

So, the probability =  \frac{4}{16}

Can you please point to where i am wrong ?

Bunuel · Answer

HHHT and TTTH does not work because in these cases the game ends on third throw not on the fourth.

Hope it helps.

vivekgautam1 · Answer

awesome !
i missed that.

Thanks Bunuel.

tejyr · Answer

Hi Experts,

I agree that the favorable outcomes are htttt or thhhh and total is 16.
I have done favorable calculation as (4!/3!)*2 and d/r is 16 which comes as 8/16.
I was wondering where I went wrong.

Bunuel · Answer

The game ends when three consecutive heads or three consecutive tails appears. So, the game to end on the fourth throw, we should have first T and then three H's or first H and then three T's. Thus, there are only two favorable cases THHH and HTTT (no need for factorial correction which accounts for different arrangements).

tejyr · Answer

Got it Thank you Bunuel for clarifying

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

M15-31

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards