Chapter 14.FOM, Problem 1P

To determine

a)

To find:

The probability of winning by using the strategy.

Answer to Problem 1P

Solution:

The probability of winning by using the strategy is $0.1$.

Explanation of Solution

Approach:

The formula of probability of an event is given by,

$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$

Here, $n\left(E\right)$ is the number of favorable outcomes for event $E$ and $n\left(S\right)$ is the number of total possible outcomes.

Calculation:

The game was played 30 times so possible chances of winning is 30.

$n\left(S\right)=30$

The number of times person won is$3$.

$n\left(E\right)=3$

Substitute $30$ for $n\left(S\right)$ and$3$ for $n\left(E\right)$ in above mentioned formula of probability.

$\begin{array}{rll}P\left(E\right)& =& \frac{3}{30}\\ & =& \frac{1}{10}\\ & =& 0.1\end{array}$

Conclusion:

Hence, the probability of winning by using the strategy is $0.1$.

To determine

b)

To find:

The probability of winning by using the switching strategy.

Answer to Problem 1P

Solution:

The probability of winning by using the switching strategy is $0.9$.

Explanation of Solution

Approach:

The formula of probability of an event is given by,

$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$

Here, $n\left(E\right)$ is the number of favorable outcomes for event $E$ and $n\left(S\right)$ is the number of total possible outcomes.

The formula for the complement of $E$ is $P\left(E^{\prime }\right)=1-P\left(E\right)$ …$\left(1\right)$

Calculation:

From part $\left(a\right)$, $P\left(E\right)=0.1$.

If the contestant decides to switch, she will switch to the winning door if she had initially chosen a losing one and vice-versa.

Substitute $0.1$ for $P\left(E\right)$ in equation $\left(1\right)$.

$\begin{array}{rll}P\left(E^{\prime }\right)& =& 1-0.1\\ & =& 0.9\end{array}$

Conclusion:

Hence, the probability of winning by using the switching strategy is $0.9$.