Question

It is estimated that 26% of all California adults are college graduates and that 31% of California adults are regular internet users. It is also estimated that 21% of California adults are both college graduates and regular internet users. (a) What is the probability that a California adult is an internet user, given that he or she is a college graduate? Round your answer to 2 decimal places. (b) Among California adults, what is the probability that a randomly chosen internet user is a college graduate? Round your answer to 2 decimal places.

Answer 1

The probability that a California adult is an internet user, given they are a college graduate, is approximately 0.81. Conversely, the probability that an internet user is a college graduate is roughly 0.68.

Step-by-step explanation:

To answer the student's questions, we first need to break down the probabilities provided:

• The probability that a California adult is a college graduate (P(College)) is 0.26.
• The probability that a California adult is a regular internet user (P(Internet)) is 0.31.
• The probability that a California adult is both a college graduate and regular internet user (P(College & Internet)) is 0.21.

(a) The probability that a California adult is an internet user, given that he or she is a college graduate, is calculated using conditional probability:

P(Internet|College) = P(College & Internet) / P(College)

P(Internet|College) = 0.21 / 0.26 ≈ 0.81

(b) The probability that a randomly chosen internet user is a college graduate:

P(College|Internet) = P(College & Internet) / P(Internet)

P(College|Internet) = 0.21 / 0.31 ≈ 0.68

Answer 2

a) There is an 81% probability that a California adult is an internet user, given that he or she is a college graduate.

b) There is a 68% probability that a randomly chosen internet user is a college graduate.

Step-by-step explanation:

The best way to solve this problem is building the Venn Diagram of these sets.

I am going to say that

A is the percentage of California adults that are college graduates.

B is the percentage of California adults that are regular internet users.

We have that:

`A = a + (A \cap B)`.

In which a are those who are only college graduates and
`(A \cap B)` are those who are both college graduates and regular internet users.

By the same logic, we have that:

`B = b + (A \cap B)`

In which b are those who are only regular internet users and
`(A \cap B)` are those who are both college graduates and regular internet users.

We start finding these values from the intersection:

It is also estimated that 21% of California adults are both college graduates and regular internet users. This means that
`A \cap B = 0.21`

26% of all California adults are college graduates. This means that
`A = 0.26`

`A = a + (A \cap B)`

`0.26 = a + 0.21`

`a = 0.05`

31% of California adults are regular internet users. This means that
`B = 0.31`

`B = b + (A \cap B)`

`0.31 = b + 0.21`

`b = 0.10`

(a) What is the probability that a California adult is an internet user, given that he or she is a college graduate?

The set of college graduates and regular internet users is given by
`A \cap B`.

The set of college graduates is given by
`A`.

So

`P = (0.21)/(0.26) = 0.81`

There is an 81% probability that a California adult is an internet user, given that he or she is a college graduate.

b) Among California adults, what is the probability that a randomly chosen internet user is a college graduate?

The set of college graduates and regular internet users is given by
`A \cap B`.

The set of internet users in given by
`B`.

So

`P = (0.21)/(0.31) = 0.68`

There is a 68% probability that a randomly chosen internet user is a college graduate.