Q. 2.4 - A First Course in Probability

Question

A, B, and C take turns flipping a coin. The first one to get a head wins. The sample space of this experiment can be defined by

$S = \left\{\begin{array}{l}1, 01, 001, 0001, . . . , \\ 0000 · · ·\end{array}\right.$

(a) Interpret the sample space.

(b) Define the following events in terms of S:

(i) $A$ wins = $A$.

(ii) $B$ wins = $B$.

(iii) ${(A\cup B)}^c$.

Assume that A flips first, then B, then C, then A, and so on.

Step-by-Step Solution

Answer

(a) The sample space tells about the winning of A, B and C.

(b) 

(i) $S_A=\left\{\left(1\right), \left(0,0,0,1\right), \left(0,0,0,0,0,0,1\right), .........\right\}$

(ii) $S_B=\left\{\left(0,1\right), \left(0,0,0,0,1\right), \left(0,0,0,0,0,0,0,1\right), .........\right\}$

(iii) $S_{{\left(A\cup B\right)}^c}=\left\{\left(0,0,1\right), \left(0,0,0,0,0,1\right), \left(0,0,0,0,0,0,0,0,1\right), .........\right\}$

Part (a) Step 1. Interpret the sample space.

It is given that, A, B, and C take turns flipping a coin. The first one to get a head wins.

The given sample space is 

$S = \left\{\begin{array}{l}1, 01, 001, 0001, . . . , \\ 0000 · · ·\end{array}\right.$

$1$: A wins on first throw.

$01$: A does not get head, B gets head and win.

$001$: A and B do not get head, C gets head and win.

$0001$: A, B, and C do not get head, A gets head in fourth toss and win.

$0000...$: A, B, and C do not get head in repeated throws.

Part (b) (i) Step 1. Define S A .

A can win in first throw or in fourth throw or in seventh throw and so on.

Therefore, $S_A=\left\{\left(1\right), \left(0,0,0,1\right), \left(0,0,0,0,0,0,1\right), .........\right\}$.

Part (b) (ii) Step 1. Define S B

B can win in second throw or in fifth throw or in eighth throw and so on. 

Therefore, $S_B=\left\{\left(0,1\right), \left(0,0,0,0,1\right), \left(0,0,0,0,0,0,0,1\right), .........\right\}$

Part (b) (iii) Step 1. Define S A ∪ B c .

$S_A=\left\{\left(1\right), \left(0,0,0,1\right), \left(0,0,0,0,0,0,1\right), .........\right\}$

$S_B=\left\{\left(0,1\right), \left(0,0,0,0,1\right), \left(0,0,0,0,0,0,0,1\right), .........\right\}$

$S_{{\left(A\cup B\right)}^c}=$A or B do not win.

Therefore, $S_{{\left(A\cup B\right)}^c}=\left\{\left(0,0,1\right), \left(0,0,0,0,0,1\right), \left(0,0,0,0,0,0,0,0,1\right), .........\right\}$