Question

A certain type of tomato seed germinates 90% of the time. A gardener planted 25 seeds. a What is the probability that exactly 20 seeds germinate? b What is the probability that 20 or more seeds germinate? c What is the probability that 24 or fewer seeds germinate? d What is the expected number of seeds that germinate?

Answer 1

Number of seeds planted : n=25

Probability that a seed germinates : p =90/100=0.9

q =1-p =0.1

X: Number of seeds germinate

X follows Binomial distribution with n=25 and p=0.9;

Probability mass function of X : Probability that 'r' seeds germinate is given by

a.

Probability that exactly 20 seeds germinate = P(X=20)

Probability that exactly 20 seeds germinate = 0.06459368

b.

probability that 20 or more seeds germinate = P(X20) = P(X=20) + P(X=21)+P(X=22)+P(X=23)+P(X=24)+P(X=25)

P(X=20) = 0.06459368

P(X20) =

P(X=20) + P(X=21)+P(X=22)+P(X=23)+P(X=24)+P(X=25)

=0.06459368+0.13841502+0.22649731+0.26588814+0.19941611+0.07178980=0.96660006

probability that 20 or more seeds germinate = 0.96660006

c. probability that 24 or fewer seeds germinate = P(X24) = 1-P(X=25)

P(X=25) = 0.07178980

P(X24) = 1-P(X=25) = 1-0.07178980=0.9282102

probability that 24 or fewer seeds germinate =0.9282102

d.

expected number of seeds that germinate

Expected value of Binomial distribution : E(X) =np

expected number of seeds that germinate 25 x 0.9 = 22.5