Inspecting switches A shipment contains $10,000$switches. Of these, $1000$ are bad. An inspector draws $2$ switches at random, one after the other.

(a) Draw a tree diagram that shows the sample space of this chance process.

(b) Find the probability that both switches are defective.

A single shipment has a total of $10,000$ switches. And there are $1000$faulty switches.

A tree diagram can be used to depict the sample space when chance behavior involves a series of outcomes. Tree diagrams can also be used to determine the likelihood of two or more events occurring at the same time. Simply multiplying along the branches that correspond to the desired results is all that is required.

There are two choices, therefore at each knot, two branches are needed:

The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:

To get the probability that both switches are defective, multiply the corresponding probabilities.

Therefore, $P(BadandBad)=P\left(Bad\right)+P(BadafterBad)$

$=\frac{1000}{10000}\times \frac{999}{9999}=\frac{999}{99990}=0.009991$

As a result, there's a good chance that both switches are bad.

$P(BadandBad)=0.009991$