Question

Cell phonesThe Pew Research Center asked a random sample of $2024$adult cell-phone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data.

Suppose we select one of the survey respondents at random. What’s the probability that:

a. The person is not age $18$to $34$and does not own an iPhone?

b. The person is age $18$to $34$or owns an iPhone?

Short answer

a. The probability of a person who is not between the ages of $18$and $34$and does not own an iPhone is approximately $0.5973.$

b. The probability of someone between the ages of $18$and $34$owning an iPhone is approximately $0.4027$

Step-by-step solution

Part (a) Step 1 : Given Information

We have to determine the probability for the person not aged 18 − 34 and does not own an iPhone.

Part (a) Step 2 : Simplification

A total of $2024$adult cell phone users are shown in the table.

Thus,

$2024$is the maximum number of possible outcomes.

Now,

The following are the people who are not between the ages of $18$and$34$ and do not own an iPhone:
Android users (other than those between the ages of $18$and$34$):
Android is used by $189$individuals aged $35$to $54$.

Android is used by $100$persons aged$55$and up.
Other cell phone users (not under the age of$18$):
Other cell phones are used by $277$adults aged $35$to $54$.

Other cell phones are used by $643$individuals aged $55$and up.

When all of the above counts are added together, we get $1209$.

That is to say,

adults between the ages of $18$and $34$do not have an iPhone.

Thus,

The total number of positive outcomes is $1209$.

The probability is calculated by dividing the number of favourable outcomes by the total number of possible possibilities.

$$\begin{gathered} P(Not\hspace{0.17em}18\hspace{0.17em}-\hspace{0.17em}34\hspace{0.17em}and\hspace{0.17em}do\hspace{0.17em}not\hspace{0.17em}own\hspace{0.17em}iPhone)=\frac{Number\hspace{0.17em}of\hspace{0.17em}favorable\hspace{0.17em}outcomes}{Number\hspace{0.17em}of\hspace{0.17em}possible\hspace{0.17em}outcomes\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\frac{1209}{2024\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\hspace{0.17em}\approx 0.5973 \end{gathered}$$
As a result, the probability of a person who is not between the ages of $18$and $34$and does not own an iPhone is approximately $0.5973.$

Part (b) Step 1 : Given Information

We have to determine the probability for the person aged $18-34$or owns an iPhone.

Part (b) Step 2 : Simplification

A total of $2024$adult cell phone users are shown in the table.

Thus,

$2024$is the maximum number of possible outcomes.

Now,

For those between the ages of $18$and$34$who own an iPhone:

$18$to $34$years old:

$517$of the $2024$adults are between the ages of $18$and $34$.

iPhone users (other than those between the ages of $18$and $34$):

iPhone is used by $171$individuals aged $35$to $54$.

iPhone is used by $127$individuals aged $55$and up.

When all of the above counts are added together, the total is $815$.

That is to say,

$815$adults between the ages of $18$and $34$own an iPhone.

Thus,

The total number of positive outcomes is$815$.

The probability is calculated by dividing the number of favourable outcomes by the total number of possible possibilities.

$$\begin{gathered} P(Aged\hspace{0.17em}18\hspace{0.17em}-\hspace{0.17em}34\hspace{0.17em}or\hspace{0.17em}own\hspace{0.17em}iPhone)=\frac{Number\hspace{0.17em}of\hspace{0.17em}favorable\hspace{0.17em}outcomes}{Number\hspace{0.17em}of\hspace{0.17em}possible\hspace{0.17em}outcomes\hspace{0.17em}} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\frac{805}{2024\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\approx 0.4027 \end{gathered}$$

Thus,

The probability of someone between the ages of $18$and $34$owning an iPhone is approximately $0.4027$.