Question

A certain type of tomato seed germinates 90% of the time. Suppose a backyard farmer planted 25 seeds

. a. What is the probability that exactly 20 germinate?

b. What is the probability that 20 or more germinate?

c. What is the probability that 24 or fewer germinate?

d. What is the expected number of seeds that germinate?

Answer 1

Let $X=$ the number of seeds that germinate: $X\sim Bin(n, p).$

Given $n=25, p=0.9, q=1-p=0.1.$

a.

$$P(X=20)=\dbinom{25}{20}(0.9)^{20}(0.1)^{25-20}$$

$$=0.06459367658$$

b.

$$P(X\geq20)=P(X=20)+P(X=21)$$

$$+P(X=22)+P(X=23)+P(X=24)$$

$$+P(X=25)=\dbinom{25}{20}(0.9)^{20}(0.1)^{25-20}$$

$$+\dbinom{25}{21}(0.9)^{21}(0.1)^{25-21}+\dbinom{25}{22}(0.9)^{22}(0.1)^{25-22}$$

$$+\dbinom{25}{23}(0.9)^{23}(0.1)^{25-23}+\dbinom{25}{24}(0.9)^{24}(0.1)^{25-24}$$

$$+\dbinom{25}{25}(0.9)^{25}(0.1)^{25-25}=0.96660005538$$

c.

$$P(X\leq24)=1-P(X=25)$$

$$=1-\dbinom{25}{25}(0.9)^{25}(0.1)^{25-25}=0.92821020123$$

d.

$$E[X]=np=25(0.9)=22.5$$