Chapter 11, Problem 19E

Solution

To explain: The chance of getting a tile with the number 5 on it among 20 tiles with numbers 1 through 20 in 20 random trails.

The probability of getting a tile with the number 5 on it among 20 tiles with numbers 1 through 20 in 20 random trails is 0.3774, so there is a chance to get a 5 once in 20 trails.

Given:

The total number of tiles $n=20$

The theoretical probability that a tile chosen at random is a 5 is $p=\frac{1}{20}$

Formula used:

The probability for x success among n independent trials where probability of each success is p is calculated using binomial probability distribution as shown below

  $\begin{array}{l}$ P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{(1-p)}^{n-x}$\left(\begin{array}{c}n\\ x\end{array}\right)=\frac{n!}{x!\left(n-x\right)!}\end{array}$

Calculation:

The probability that exactly once a tile with the number 5 is picked in 20 trails is calculated as shown below

  $\begin{array}{l}$ P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{(1-p)}^{n-x}$$ P(X=1)=\left(\begin{array}{c}20\\ 1\end{array}\right){\left(\frac{1}{20}\right)}^1{\left(1-\frac{1}{20}\right)}^{20-1}$$ P(X=1)=\frac{20!}{1!\left(20-1\right)!}\times {0.05}^1\times {0.95}^{19}$P(X=1)=0.377354\end{array}$

The probability that exactly once a tile with the number 5 is picked in 20 trails is 0.3774 (rounded to 4 decimal places).

Interpretation:

The probability that tile with the number 5 is occurred exactly once in 20 trials is 0.3774. Thus it can be said that there is a chance that a tile with the number 5 can be selected once in 20 trials.