Chapter 5.3, Problem 86E

To determine

Probability for the randomly chosen object from the beach is a metal object.

Answer to Problem 86E

Probability that the boy has found a metal object is approx. 0.8909.

Explanation of Solution

Given information:

Metal detector data when it passes over the objects (metal or non − metal):

Machine beeps for 98% of the metal objects over which it passes.

Machine beeps for 4% of the nonmetallic objects over which it passes.

Machine passes over 25% of the metal objects.

Calculations:

According to complement rule,

  $P\left({\text{A}}^{\text{c}}\right)=P\left(\text{not}\text{A}\right)=1-P\left(\text{A}\right)$

According to the general multiplication rule,

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

According to the addition rule for mutually exclusive events,

  $P\left(\text{A}\cup \text{B}\right)=P\left(\text{A}\text{or}\text{}\text{B}\right)=P\left(\text{A}\right)+P\left(\text{B}\right)$

Definition for conditional probability:

  $P\left(\text{B}|\text{A}\right)=\frac{P\left(\text{A}\cap \text{B}\right)}{P\left(\text{A}\right)}=\frac{P\left(\text{A}\text{and}\text{B}\right)}{P\left(\text{A}\right)}$

Let

M: Metal

M^c: Non − metal

B: Machine beeps

B^c: Machine does not beep

Now,

The corresponding probabilities:

Probability for getting metal,

  $P\left(\text{M}\right)=0.25$

Probability for getting metal when machine beeps,

  $P\left(\text{B}|\text{M}\right)=0.98$

Probability for getting non − metal when machine beeps,

  $P\left(\text{B}|{\text{M}}^{\text{c}}\right)=0.04$

Now,

Apply the complement rule:

Probability for getting non − metal,

  $P\left({\text{M}}^{\text{c}}\right)=1-P\left(\text{M}\right)=1-0.25=0.75$

Then

Apply general multiplication rule:

Probability for machine beeps and getting metal,

  $P\left(\text{B}\text{and}\text{M}\right)=P\left(\text{M}\right)\times P\left(\text{B}|\text{M}\right)=0.25\times 0.98=0.245$

Probability for machine beeps and getting non − metal,

  $P\left(\text{B}\text{and}{\text{M}}^{\text{c}}\right)=P\left({\text{M}}^{\text{c}}\right)\times P\left(\text{B}|{\text{M}}^{\text{c}}\right)=0.75\times 0.04=0.030$

We know that

Machine passes over either metal or non - metal.

Since both events are not possible at same time,

Apply addition rule for mutually exclusive events:

Probability for machine beeps,

  $\begin{array}{l}P\left(\text{B}\right)=P\left(\text{B}\text{and}\text{M}\right)+P\left(\text{B}\text{and}{\text{M}}^{\text{c}}\right)\\ =0.245+0.030\\ =0.275\end{array}$

Using conditional probability definition:

  $P\left(\text{M}|\text{B}\right)=\frac{P\left(\text{B}\text{and}\text{M}\right)}{P\left(\text{B}\right)}=\frac{0.245}{0.275}=\frac{49}{55}\approx 0.8909$

Thus,

If the machine beeps while passing over randomly selected object, the probability of getting a metal object is approx. 0.8909.