Question

The chart shows a certain city's population by age. Assume that the selections are independent events. If 8 residents of this city are selected at random, find the probability that the first 2 are 65 or older, the next 3 are 25-44 years old, the next 2 are 24 or younger, and the last is 45-64 years old.

Answer 1

0.000014

Explanation:

The chart is not provided so i will use an example chart to explain the answer. Here is a sample chart:

City X's Population by Age

0-24 years old 33%

25-44 years old 22%

45-64 years old 21%

65 or older 24%

In order to find probability of independent events we find the probability of each event occurring separately and then multiply the calculated probabilities together in the following way:

P(A and B) = P(A) * P(B)

probability that the first 2 are 65 or older

Let A be the event that the first 2 are 65 or older

The probability of 65 or older 24% i.e. 0.24

So the probability that first 2 are 65 or older is:

0.24(select resident 1) * 0.24(select resident 2)

P(A) = 0.24 * 0.24

= 0.0576

P(A) = 0.0576

probability that the next 3 are 25-44 years old

Let B be the event that the next 3 are 25-44 years old

25-44 years old 22% i.e. 0.22

So the probability that the next 3 are 25-44 years old is:

0.22 * 0.22* 0.22

P(B) = 0.22 * 0.22 * 0.22

= 0.010648

P(B) = 0.010648

probability that next 2 are 24 or younger

Let C be the event that the next 2 are 24 or younger

0-24 years old 33% i.e. 0.33

So the probability that the next 2 are 24 or younger is:

0.33 * 0.33

P(C) = 0.33 * 0.33

= 0.1089

P(C) = 0.1089

probability that last is 45-64 years old

Let D be the event that last is 45-64 years old

45-64 years old 21% i.e. 0.21

So the probability that last is 45-64 years old is:

0.21

P(D) = 0.21

So probability of these independent events is computed as:

P(A and B and C and D) = P(A) * P(B) * P(C) * P(C)

= 0.0576 * 0.010648 * 0.1089 * 0.21

= 0.000014