# Problem 421

About 14% of the population of a large country is computer illiterate. If two people are randomly selected, what is the probability both are computer illiterate? What is the probability at least one is computer illiterate?

Solution

In this problem you are given the fact that 14% of the population of a large country is computer illiterate. The first step is to make this percentage into probability.

To convert a percent into a probability, remove the % sign and then divide by 100.

The probability that a random person in the population is computer illiterate is 0.14.

(a)    Now that you know the probability of one person being computer illiterate, you can find the answer to the first part of the problem.

Let the event E be “person A is computer illiterate” and the event F be “person B is computer illiterate.” The events E and F are independent because the probability of one event is not effected by the outcome of the other event.

Both P(E) = 0.14 and P(F) = 0.14.

Since the event E and F are independent, you can the multiplication rule, which says that if E and F are independent events, than P(E and F) = P(E) * P(F).

P(E and F) = P(E) * P(F) = 0.0196

(b)   The phrase “at least” means “greater than or equal to”, so you wish to known the probability that only one or both people are computer illiterate. These events are mutually exclusive.

Since the events that only one or both people are computer illiterate are mutually exclusive, you can the addition rule for disjoint events to compute that at least one person is computer illiterate.

P(only 1 or both) = P(only 1) +P (both)

Generally, computing these probabilities is very time consuming. However, notice that the complement of “at least one person is computer illiterate” is “neither person is computer illiterate.” You should use the complement rule to compute the probability.

P(at least 1 computer illiterate) = 1 – P(neither computer illiterate)

Now , what is the probability that neither of the two people are computer illiterate? To find this , first find the probability that one is not computer illiterate. Recall, for probability P(E),

P(E) = 1 – P(E)

The probability that  1 person is not computer illiterate is 0.86.

To find the probability that both people are  not computer illiterate, you use the multiplication rule, which says that if E and F are independent events, than P(E and F) = P(E) * P(F).

P(neither E) = P(person 1 not E) * P(person 2 not  E)

= 0.86 * 0.86

= 0.7396

Finally , compute the probability that at least one person is computer illiterate.

P(at least 1 computer illiterate) = 1 – P(neither computer illiterate)

= 0.2604