Question

According to a study done by a university​ student, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe​ people's habits as they sneeze.
​(a) What is the probability that among 16 randomly observed individuals exactly 8 do not cover their mouth when​ sneezing?
​(b) What is the probability that among 16 randomly observed individuals fewer than 4 do not cover their mouth when​ sneezing?
​(c) Would you be surprised​ if, after observing 16 ​individuals, fewer than half covered their mouth when​ sneezing? Why?

Answer 1

Answer:

a.  P(X=8)=0.0277

b.  P(X<4)=0.3460

c. No. P(X<8)=0.9605

Step-by-step explanation:

a. Let x denote the event.

This is a binomial probability distribution problem expressed as

$P(X=x)={n\choose x}p^x(1-p)^{n-x}$

Where

• n is the total number of events
• p is the probability of a success
• x is the number of successful events.

Given that n=16, p=0.267, the probability of exactly 8 people not covering their mouths is calculated as:

$P(X=x)={n\choose x}p^x(1-p)^{n-x}\\\\\\P(X=8)={16\choose 8}0.267^8(1-0.267)^8\\\\\\=0.0277$

Hence, the probability of exactly 8 people not covering their mouths is 0.0277

b. The probability of fewer than 4 people covering their mouths is calculated as:

-We calculate and sum the probabilities of exactly 0 to exactly 3:

$P(X=x)={n\choose x}p^x(1-p)^{n-x}\\\\P(X$

Hence, the probability of x<4 is 0.3460

c. Would you be surprised if fewer than half covered their mouths:

The probability  of fewer than half covering their mouths is calculated as:

$P(X=x)={n\choose x}p^x(1-p)^{n-x}\\\\P(X$

No. The probability of fewer than half is 0.9605 or 96.05%. This a particularly high probability that erases any chance of doubt or surprise.