Answer :
Let's list down the given information the question.
Probability of success p (smartphone users) = 54%
probability of failure q (non-smartphone users) = 100% - 54% = 46%
n = 7
To be able to get the probability of at least 5 out 7 uses smartphones in the meetings, we will have to solve the probability of getting exactly 5, 6, and 7 smartphone users and add the results.
The formula to use is:
[tex]P(x)=nCx\times p^x\times q^{n-x}[/tex]where p, q, and n were already defined above and x = the number of success.
Let's begin calculating the probability of x = 5 smartphone users. Let's plug in the values of n, x, p, and q in the formula above.
[tex]\begin{gathered} P(5)=7C5\times0.54^5\times0.46^2 \\ P(5)=21\times0.0459165\times0.2116 \\ P(5)=0.2040346 \end{gathered}[/tex]The probability of getting exactly 5 smartphone users is 0.2040346.
Let's solve for the probability of x = 6 smartphone users. Plugging in the values of n, x, p, and q, we get:
[tex]\begin{gathered} P(6)=7C6\times0.54^6\times0.46^1 \\ P(6)=7\times0.0247949\times0.46 \\ P(6)=0.079839 \end{gathered}[/tex]The probability of getting exactly 6 smartphone users is 0.079839.
Lastly, let's solve for the probability of x = 7 smartphone users. Plugging in the values of n, x, p, and q, we get:
[tex]\begin{gathered} P(7)=7C7\times0.54^7\times0.46^0 \\ P(7)=1\times0.013389\times1 \\ P(7)=0.013389 \end{gathered}[/tex]The probability of getting exactly 7 smartphone users is 0.013389.
As mentioned, to get the probability of at least 5 out 7 uses smartphones in the meetings, we will have to solve the probability of getting exactly 5, 6, and 7 smartphone users and add the results.
[tex]\begin{gathered} P(x\ge5)=P(5)+P(6)+P(7) \\ P(x\ge5)=0.2040346+0.079839+0.013389_{} \\ P(x\ge5)=0.2973 \end{gathered}[/tex]Therefore, the probability of at least 5 out 7 randomly uses smartphones in the meeting is 0.2973.
