Question

write the binomial probability and the normal probability for the shaded region of the graph. find the value of each probability and compare the results.
write the binomial probability for the shaded region of the graph and find its value. select the correct choice below and fill in the answer box within your choice.
(round to four decimal places as needed.)
○ a. ( p(4<x<8)=p(4.5)+p(5.5)+p(6.5)+p(7.5)=)
○ b. ( p(4 leq x leq 8)=p(4)+p(5)+p(6)+p(7)+p(8)=)
○ c. ( p(5 leq x leq 7)=p(5)+p(6)+p(7)=)

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\), \(n = 16\), \(p=0.5\), \(1-p = 0.5\)

Step2: Calculate \(P(5)\)

For \(k = 5\):
\(C(16,5)=\frac{16!}{5!(16 - 5)!}=\frac{16\times15\times14\times13\times12}{5\times4\times3\times2\times1}=4368\)
\(P(5)=C(16,5)\times(0.5)^{5}\times(0.5)^{16 - 5}=4368\times(0.5)^{16}\)
\(P(5)=4368\times\frac{1}{65536}\approx0.0666\)

Step3: Calculate \(P(6)\)

For \(k = 6\):
\(C(16,6)=\frac{16!}{6!(16 - 6)!}=\frac{16\times15\times14\times13\times12\times11}{6\times5\times4\times3\times2\times1}=8008\)
\(P(6)=C(16,6)\times(0.5)^{6}\times(0.5)^{16 - 6}=8008\times(0.5)^{16}\)
\(P(6)=8008\times\frac{1}{65536}\approx0.1222\)

Step4: Calculate \(P(7)\)

For \(k = 7\):
\(C(16,7)=\frac{16!}{7!(16 - 7)!}=\frac{16\times15\times14\times13\times12\times11\times10}{7\times6\times5\times4\times3\times2\times1}=11440\)
\(P(7)=C(16,7)\times(0.5)^{7}\times(0.5)^{16 - 7}=11440\times(0.5)^{16}\)
\(P(7)=11440\times\frac{1}{65536}\approx0.1746\)

Step5: Sum the probabilities

\(P(5\leq x\leq7)=P(5)+P(6)+P(7)\)
\(P(5\leq x\leq7)=0.0666 + 0.1222+0.1746=0.3634\)

Answer:

C. \(P(5\leq x\leq7)=P(5)+P(6)+P(7)=0.3634\)