Answer:
a. Â P(X=8)=0.0277
b. Â 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
[tex]P(X=x)={n\choose x}p^x(1-p)^{n-x}[/tex]
Where
Given that n=16, p=0.267, the probability of exactly 8 people not covering their mouths is calculated as:
[tex]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[/tex]
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:
[tex]P(X=x)={n\choose x}p^x(1-p)^{n-x}\\\\P(X<4)=P(X=0)+P(X=1)+P(X=2)+P(X=3)\\\\={16\choose 0}0.267^0(0.733)^{16}+{16\choose 1}0.267^0(0.733)^{15}+{16\choose 2}0.267^2(0.733)^{14}+{16\choose 3}0.267^3(0.733)^{13}\\\\=0.0069+0.0405+0.1106+0.1880\\\\=0.3460[/tex]
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:
[tex]P(X=x)={n\choose x}p^x(1-p)^{n-x}\\\\P(X<8)=P(X<4)+P(X=4)+P(X=5)+P(X=6)+P(X=7)\\\\=0.3460+{16\choose 4}0.267^4(0.733)^{12}+{16\choose 5}0.267^5(0.733)^{11}+{16\choose 6}0.267^6(0.733)^{10}+{16\choose 7}0.267^7(0.733)^{9}\\\\=0.3460+0.2225+0.1945+0.1299+0.0676\\\\=0.9605[/tex]
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.
