Chapter 9.7, Problem 60E

a.

To determine

To find: the probability that both vehicles are available at a given time.

Answer to Problem 60E

  $0.81$

Explanation of Solution

Concept Used:

Let A and B be two independent events, then

  $P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)$

Let E be the event that both vehicles are available at a given time.

Here, availability of one vehicle is independent of the availability of the other vehicle.

Also, the probability that a specific vehicle is available when needed is 90%.

Let A denotes the event that 1^st rescue vehicle is available.

And let B denotes the event that 2^nd rescue vehicle is available.

Then,

  $P\left(A\right)=P\left(B\right)=0.90$

So, the probability that both vehicles are available at a given time is:

  $\begin{array}{c}P\left(E\right)=P\left(\text{A and }B\right)\\ =P\left(A\right)\times P\left(B\right)\\ =0.90\times 0.90\\ =0.81\end{array}$

b.

To determine

To find: the probability that neither vehicle is available at a given time.

Answer to Problem 60E

  $0.01$

Explanation of Solution

Concept Used:

Let A and B be two independent events, then

  $P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)$

Let $E^{\prime }$ be the event that neither vehicle is available at a given time.

Here, availability of one vehicle is independent of the availability of the other vehicle.

Also, the probability that a specific vehicle is available when needed is 90%.

So, the probability that a specific vehicle is not available when needed is $10\%$ .

Let $A^{\prime }$ denotes the event that 1^st rescue vehicle is not available.

And let $B^{\prime }$ denotes the event that 2^nd rescue vehicle is not available.

Then,

  $P\left(A^{\prime }\right)=P\left(B^{\prime }\right)=0.10$

So, the probability that neither vehicle is available at a given time is:

  $\begin{array}{c}P\left(E^{\prime }\right)=P\left(A^{\prime }\text{ and }{B}^{\prime }\right)\\ =P\left(A^{\prime }\right)\times P\left(B^{\prime }\right)\\ =0.10\times 0.10\\ =0.01\end{array}$

c.

To determine

To find: the probability that at least one vehicle is available at a given time.

Answer to Problem 60E

  $0.99$

Explanation of Solution

Concept Used:

Let A and B be two independent events, then

  $P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)$

Let F be the event that at least one vehicle is available at a given time.

Event F is complement of the event $E^{\prime }$ of part (b) that neither vehicle is available at a given time.

So,

  $P\left(F\right)=1-P\left(E^{\prime }\right)$

From part (b), $P\left(E^{\prime }\right)=0.01$

So, the probability that at least one vehicle is available at a given time is:

  $\begin{array}{c}P\left(F\right)=1-0.01\\ =0.99\end{array}$