Question

a company tracks its apps user engagement and reports that 25% of their users engage daily. suppose that 11 users are selected randomly.
a. is this selection of the 11 users a binomial experiment? explain your answer.
since the users are selected randomly, the probability, p, does not change from trial to trial, the trials are independent, and there are 2 outcomes for each trial, the selection is a binomial experiment.
b. find the probability that all selected users engage daily. (round your answer to four decimal places.)
c. find the probability that exactly 3 of the selected users engage daily. (round your answer to four decimal places.)
d. find the probability that at least one of the selected users engages daily. (round your answer to four decimal places.)

Explanation:

Step1: Determine if it is a binomial experiment

A binomial experiment has the following properties:

• Fixed number of trials (\(n = 11\) users selected)
• Each trial has two outcomes (user engages daily or not)
• Probability of success (\(p=0.25\)) remains the same for each trial
• Trials are independent (users are selected randomly)

Since all these properties are met, it is a binomial experiment.

Step2: Use the binomial probability formula \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\)

For part b: \(n = 11\), \(k = 11\), \(p = 0.25\), \(1-p=0.75\)
\(C(11,11)=\frac{11!}{11!(11 - 11)!}=1\)
\(P(X = 11)=1\times(0.25)^{11}\times(0.75)^{0}\approx0\)

For part c: \(n = 11\), \(k = 3\)
\(C(11,3)=\frac{11!}{3!(11 - 3)!}=\frac{11\times10\times9}{3\times2\times1}=165\)
\(P(X = 3)=165\times(0.25)^{3}\times(0.75)^{8}\)
\(=165\times0.015625\times0.100112915\approx0.2581\)

For part d: Use the complement rule \(P(X\geq1)=1 - P(X = 0)\)
\(C(11,0)=\frac{11!}{0!(11-0)!}=1\)
\(P(X = 0)=1\times(0.25)^{0}\times(0.75)^{11}\)
\(=(0.75)^{11}\approx0.0422\)
\(P(X\geq1)=1 - 0.0422 = 0.9578\)

Answer:

a. Yes, it is a binomial experiment because there is a fixed number of independent trials (\(n = 11\)), two outcomes per trial (engage or not engage), constant probability of success (\(p=0.25\)).
b. \(0\)
c. \(0.2581\)
d. \(0.9578\)