Chapter 17, Problem 9E

(a)

To determine

To find: the probability of this researchers losing their 5th attempt for the first time.

Answer to Problem 9E

0.0819

Explanation of Solution

Given:

X is the number of attempts that succeed in producing the first bad shot.

probability of success is $p=0.2$ .

Formula used:

  $P\left(X=x\right)=p.q^{x-1}$

Calculation:

And

  $\begin{array}{c}q=1-p\\ =1-0.2\\ =0.8\end{array}$

The probability mass function for the geometric distribution is.

  $\begin{array}{c}P\left(X=x\right)=q^{x-1}p\\ ={\left(0.8\right)}^{x-1}\left(0.2\right)\end{array}$

Required probability

  $\begin{array}{c}P\left(X=5\right)={\left(0.8\right)}^4\left(0.2\right)\\ =0.08192\\ =0.0819\end{array}$

(b)

To determine

To find: In this game researchers are likely to hit their 1^st basket on their 4^th attempt.

Answer to Problem 9E

0.0064

Explanation of Solution

Given:

X is the number of attempts that succeed in producing the first bad shot.

probability of success is $p=0.2$

Formula used:

  $P\left(X=x\right)=p.q^{x-1}$

Calculation:

the possibility of his fourth shot in the first basket.

  $\begin{array}{c}P\left(X=4\right)={\left(0.2\right)}^3\left(0.8\right)\\ =0.0064\end{array}$

There is 0.64% probability that his 1^st basket is his 4^th shot.

(c)

To determine

To find: The probability of the first basket hitting his first 3 shots in this game researcher.

Answer to Problem 9E

0.992

Explanation of Solution

Given:

X is the number of attempts that succeed in producing the first bad shot.

probability of success is $p=0.2$

Formula used:

  $P\left(X=x\right)=p.q^{x-1}$

Calculation:

Find the possibility researcher 's taken one of his first 3 shots for his first basket.

Thus, the required probability is,

  $\begin{array}{c}P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right)=0.8+\left(0.2\right)\left(0.8\right)+{\left(0.2\right)}^2\left(0.8\right)\\ =0.8+0.16+0.032\\ =0.992\end{array}$