Question

A survey found that 73% of adults have a landline at their residence (event A); 83% have a cell phone (event B). It is known that 2% of adults have neither a cell phone nor a landline. 3. What is the probability that an adult selected at random has both a landline and a cell phone? A. 0.58 B. 0.98 C. 0.6059 D. None of these Work: 4. Given an adult has a cell phone, what is the probability he does not have a landline?

A. 0.27
B. 0.25
C. 0.3012
D. None of these

Answer 1

3. What is the probability that an adult selected at random has both a landline and a cell phone?

A. 0.58

4. Given an adult has a cell phone, what is the probability he does not have a landline?

C. 0.3012

Explanation:

We solve this problem building the Venn's diagram of these probabilities.

I am going to say that:

A is the probability that an adult has a landline at his residence.

B is the probability that an adult has a cell phone.

C is the probability that a mean is neither of those.

We have that:

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

In which a is the probability that an adult has a landline but not a cell phone and
`A \cap B` is the probability that an adult has both of these things.

By the same logic, we have that:

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

The sum of all the subsets is 1:

`a + b + (A \cap B) + C = 1`

2% of adults have neither a cell phone nor a landline.

This means that
`C = 0.02`.

73% of adults have a landline at their residence (event A); 83% have a cell phone (event B)

So
`A = 0.73, B = 0.83`.

What is the probability that an adult selected at random has both a landline and a cell phone?

This is
`A \cap B`.

We have that
`A = 0.73`. So

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

`a = 0.73 - (A \cap B)`

By the same logic, we have that:

`b = 0.83 - (A \cap B)`.

So

`a + b + (A \cap B) + C = 1`

`0.73 - (A \cap B) + 0.83 - (A \cap B) + (A \cap B) + 0.02 = 1`

`(A \cap B) = 0.75 + 0.83 - 1 = 0.58`

So the answer for question 3 is A.

4. Given an adult has a cell phone, what is the probability he does not have a landline?

83% of the adults have a cellphone.

We have that

`b = B - (A \cap B) = 0.83 - 0.58 = 0.25`

25% of those do not have a landline.

So
`P = (0.25)/(0.83) = 0.3012`

The answer for question 4 is C.