Coin toss, with 2 people. First one who tossed a tails wins.

Question

Amina and Bert alternate turns tossing a fair coin. Amina goes first and each player takes three turns. The first player to toss a tail wins. If neither Amina nor Bert tosses a tail, neither wins. What is the probability of Amina winning. Show all work.
A) 21/32
B)5/8
C)3/7
D)11/16
E)9/16

Answer 1

Let's use the letters w and l to indicate whether an individual throw is a win or loss.  Amina alternates her tosses with Bert.  Keeping that in mind, the outcomes which indicate Amina winning are

 

Amina wins; Amina loses, Bert loses, Amina wins; Amina loses, Bert loses, Amina loses, Bert loses, Amina wins.  Using l and w to indicate Amina winning or losing her rounds, we have

 

{w, llw, llllw}

 

as all of the possible ways that Amina can win.  The probability that she wins is the sum of probabilities of each outcome(denoted as P[ ]). 

 

P[w] = .5

P[llw] = .5*.5*.5 = .125

P[llllw] = .5*.5*.5*.5*.5 = .03125

 

Their sum is .65625, which corresponds to answer A.

 

I hope your teacher has shown you tree diagrams!  This problem is pretty easy if you use them; hard otherwise. The interesting thing to note here is that, even though all coin flips have the same probability of coming up heads or tails, the person who goes first has an advantage (65% chance of winning).