Chapter 7, Problem 7.1P

A player throws a fair die and simultaneously flips a fair coin, If the coin lands heads, then she wins twice, and if tails, then she wins one-half of the value that appears on the die. Determine her expected winnings.

To determine

To Find :The expected winnings after throwing the fair dice and flips a fair coin.

Answer to Problem 7.1P

The expected winnings is 4.375.

Explanation of Solution

Given information:

A player throws a fair die and simultaneously flips a fair coin.

If the coin lands heads, then she wins twice, and it tails, then she wins one-half of the value that appears on the die.

Consider X is the random variable that represents the result on the toss a fair coin.

Consider Y is the random variable that represents the number on the die.

Let's roll heads, then win twice the digits on the die roll, and if die roll a tail then we win 1/2 the digit on the die.

The probability that we get Heads/ Tails is,

  $\begin{array}{cc}& P(X=heads)=P(X=Tails)=\frac{1}{2}\\ & \text{The probability that we get any of individual number on the die is,}\\ & P(Y=1)=P(Y=2)=\dots .=P(Y=6)=\frac{1}{6}\end{array}$

Firstly, Find The expected winnings if the coin lands heads.

  $\begin{array}{l}{E}_{\text{heads}}(Y\mid X=\text{Heads})=E(2Y)P(X=\text{Heads})\\ \begin{array}{l}=2P(X=Heads)E(Y)\hfill \\ =2P[X=Heads]E(Y)\hfill \\ =2\left(\frac{1}{2}\right)\left(1\times \frac{1}{6}+2\times \frac{1}{6}+\dots +6\times \frac{1}{6}\right)\hfill \\ =\left(\frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{4}{6}+\frac{5}{6}+\frac{6}{6}\right)\hfill \\ =3.5\hfill \end{array}\end{array}$

Now, to find the expected winnings of the coin lands tails,

  $\begin{array}{l}{E}_{\text{Tails}}(Y\mid X=\text{Tails})=E\left(\frac{Y}{2}\right)P(X=\text{Tails})\\ \begin{array}{l}=\frac{1}{2}E(Y)P(X=\text{Tails})\hfill \\ =\frac{1}{2}\times \frac{1}{2}\times \left(1\times \frac{1}{6}+2\times \frac{1}{6}+\dots .+6\times \frac{1}{6}\right)\hfill \\ =\frac{1}{4}(3.5)\hfill \\ =0.875\hfill \end{array}\end{array}$

Therefore, the expected winnings is,

  $\begin{array}{cc}E& =E_{\text{Heads}}(Y\mid X=heads)+E_{T\text{ails}}(Y\mid X=\text{Tails})\\ & =3.5+0.875\\ & =4.375\end{array}$