Question

Problem 5: The probability that a patient fully recovers from a severe back pain after a certain type of therapy is 0.85. Suppose 20 people are known to have severe back pain were subjected this type of therapy. (1) Find the probability that exactly 15 fully recover? (2) Find the probability that at least 9 recover? (3) Find the probability that at least 14 but not more than 18 recover? (4) Find the probability that at most 15 recover? (5) What is the expected number of those who recover? (6) What is the standard deviation of the number of those who recover?

Answer 1

# from above in

the probability that a patient fully recovers from server back pain after a certain of therapy is 0.85

#consider a random sample of 20

.let X be the random variable that represent that a patient fully recovers from server back pain after a certain of therapy

X~Binomial(n=20,p=0.85)

#1) P(x=15)

$P(X=15)=\binom{20}{15}*(0.85)^15*(1-0.85)^{20-15}$

P(x=5)=0.1028

2) at least 9 ie

$P(X\geq 9)=\sum_{x=9}^{20}\binom{20}{x}*(0.85)^x*(1-0.85)^{20-x}$

$P(X\geq 9)=0.9999$

3) at least 14 ie

$P(X\geq 14)=\sum_{x=14}^{20}\binom{20}{x}*(0.85)^x*(1-0.85)^{20-x}$

$P(X\geq 14)=0.9781$

4) atmost 15 ie

$P(X< 15)=\sum_{x=0}^{15}\binom{20}{x}*(0.85)^x*(1-0.85)^{20-x}$

$P(X\leq 15)=0.1702$

5) expected value =n*p=20*0.85=17

6) standard deviation

$\sqrt{n*p*(1-p)}$

$\sqrt{20*0.85*(1-0.85)}$

#standard deviation=1.5969