Question

In a series of $3$ one-day cricket matches between teams $A$ and $B$ of a college, the probability of team $A$ winning or drawing are $\frac{1}{3}$ and $\frac{1}{6}$ respectively.

If a win, loss or draw gives $2$, $0$ and $1$ point respectively, then what is the probability that team $A$will score $5$ points in the series?

• A. $\frac{17}{18}$
• B. $\frac{11}{12}$
• C. $\frac{1}{12}$
• D. $\frac{1}{18}$

Answer 1

The correct option is D

$\frac{1}{18}$

Explanation for the correct option.

Step 1. Find the possible scenarios.

If a team wins it gets $2$ points, if the team loses it gets $0$ points and if the match ends in a draw then they get $1$ points.

So for team A to score $5$ points in a $3$ match series they need to win twice and draw a match once.

Let $W$ be the event that team $A$wins the game, then it is given that $P\left(W\right)=\frac{1}{3}$.

Let $D$ be the event that team $A$ draws the game, then it is given that $P\left(D\right)=\frac{1}{6}$.

Then to score $5$ points the possible scenarios are: $WWD,WDW,DWW$.

Step 2. Find the probability.

As winning a game and drawing a game are independent event, so the probability can be given as:

$\begin{array}{rcl}P\left(WWD\right)+P\left(WDW\right)+P\left(DWW\right)& =& P\left(W\right)P\left(W\right)P\left(D\right)+P\left(W\right)P\left(D\right)P\left(W\right)+P\left(D\right)P\left(W\right)P\left(W\right)\\ & =& 3P\left(W\right)P\left(W\right)P\left(D\right)\\ & =& 3\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{6}\left[P\left(W\right)=\frac{1}{3},P\left(D\right)=\frac{1}{6}\right]\\ & =& \frac{1}{18}\end{array}$

So, the probability that team $A$ will score $5$ points in the series is $\frac{1}{18}$.

Hence, the correct option is (D).