Cell phones The 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.

1. Find P(iPhone | $18-34$)
2. Use your answer from part (a) to help determine if the events “iPhone” and $"18-34"$are independent.

We are given a table that summarizes data of $2024$adult cell phone owners from united states and their age:

We need to find P( iPhone |$18-34$)

P(A|B) means probability of happening of event A when event B has already taken place. Formula for same is $P(A|B)=\frac{P(A\cap B)}{P\left(B\right)}$

According to ques event A is adults having iPhone and event B is adults of age group $18-34$

$P(A\cap B)$means probability of adult having iPhone and is in age group $18-34$

localid="1653965385334" $P(A\cap B)=\frac{No.offavourableoutcomes}{No.oftotaloutcomes}=\frac{169}{2024}$

P(B) is probability of adult having phone in age group of $18-34$

$P(18-34)=\frac{No.offavouableoutcomes}{No.oftotaloutcomes}=\frac{517}{2024}$

P(iPhone | $18-34$)= $\frac{P(iPhone\cap 18-34)}{P(18-34)}=\frac{{\displaystyle \raisebox{1ex}{$169$}\!\left/ \!\raisebox{-1ex}{$\cancel{2024}$}\right.}}{{\displaystyle \raisebox{1ex}{$517$}\!\left/ \!\raisebox{-1ex}{$\cancel{2024}$}\right.}}=\frac{169}{517}$

Hence, P(iPhone | $18-34$)= $\frac{169}{517}$

We are given a table that summarizes data of $2024$adult cell phone owners from united states and their age:

From results of part(a) we need to determine if events "iPhone" and "$18-34$" are independent.

For events to be independent P(A|B)= P(A)

According to multiplication rule of independent events P(A$\cap$B)= P(A)$\times$P(B)

From part (a) step $2$we get P(iPhone $\cap$$18-34$)=role="math" localid="1653966932798" $\frac{169}{2024}=0.083$---------------($1$)

P(iPhone)= $\frac{467}{2024}$----------($2$)

P($18-34$)=$\frac{517}{2024}$-------------($3$)

Using data calculated above, we get P(iPhone)$\times$P($18-34$)=role="math" localid="1653966911398" $\frac{416\times 517}{2024\times 2024}=0.052$

As P(iPhone)$\times$P($18-34$) $\ne$P(iPhone | $18-34$)

Hence, events are not independent.