1. Movies on Phone Twenty-five percent of adults say they would watch streaming movies on their phone at work. You randomly select 12 adults and ask them if they would watch streaming movies on their phone at work. Find the probability that the number who say they would watch streaming movies on their phone at work is

(a) exactly four,
(b) more than four, and
(c) between four and eight, inclusive.

Solution

Here $X$ denote the number of adults who watch streaming movies on their phone at work. Twenty-five percent of adults say they would watch streaming movies on their phone at work. So $p= 0.25$.

Given that random variable $X$ follows Binomial distribution. That is $X\sim B(12, 0.25)$.

The probability mass function of $X$ is
$$ \begin{aligned} P(X=x) &= \binom{12}{x} (0.25)^x (1-0.25)^{12-x},\\ & \quad\; x=0,1,\cdots, 12. \end{aligned} $$

(a) The probability that $X$ is exactly $4$ is

$$ \begin{aligned} P(X= 4) & =\binom{12}{4} (0.25)^{4} (1-0.25)^{12-4}\\ & = 0.1936\\ \end{aligned} $$

(b) The probability that more than $4$ watch streaming movies on their phone at work is $P(X\geq 4)$.

$$ \begin{aligned} P(X\geq 4) & = 1- P(X\leq 3)\\ & = 1- \big(P(X=0) + P(X=1)+P(X=2) + P(X=3)\big) \\ &= 1- \big(0.0317+0.1267+0.2323+0.2581\big)\\ & = 1- 0.6488\\ & = 0.3512 \end{aligned} $$

(c) The probability that between 4 and 8, inclusive, watch streaming movies on their phone at work is $P(4\leq X\leq 8)$.

$$ \begin{aligned} P(4\leq X\leq 8) & = P(X=4) + P(X=5)+P(X=6) + P(X=7)+P(X=8) \\ &= 0.1936 +0.1032+0.0401+0.0115+0.0115\\ & = 0.3508 \end{aligned} $$