Q.4.22 - A First Course in Probability

Question

 Each game you play is a win with probability $p$. You plan to play $5$ games, but if you win the fifth game, then you will keep on playing until you lose.

1. Find the expected number of games that you play. 
2. Find the expected number of games that you lose. 

Step-by-Step Solution

Answer

1. The expected number of games that you play is $4+\frac{1}{1-p}$
2. The expected number of games that you lose is $1+4(1-p)$

Step 1:Given Information (Part -a)

Given in the question that, you plan to play a $5$ games. Each game you play is a win with probability $p$. we have to find the expected number of games that you play.

Step 2 : Explanation (Part-a)

Here, we are going to play $5$ games.

Let's consider the first four games separately. From the $5^{th}$ game and on we need to play until we lose.

Therefore, the total number of games can be indicated as $4+x$.

Where $X$ has geometric distribution with parameter of success $1-p$

Hence, the expected number of games that will be played is

 $E(4+X)=4+EX=4+\frac{1}{1-p}$

Step 3 :Final Answer (Part-a)

The expected number of games that will be played is $E\left(4+X\right)=4+\frac{1}{1-P}$

Step 4 :Given information (Part-b)

Given in the question that, you plan to play a $5$games. Each game you play is a win with probability $p$. We have to find the expected number of games that you lose.

Step 5 :Explanation (Part-b)

Consider $Y$ as the number of games that we lose.

So, $Y$ can be written as $Y=Z+1$, where $Z$ indicates the number of games that we lose within first four games.

 Hence,$E\left(Y\right)=1+E\left(Z\right)=1+4(1-p)$

Step 6:Final Answer (Part-b)

The expected number of games that you lose is$E\left(Y\right)=1+4\left(1-P\right)$