Question

According to genetic theory, the blossom color in the second generation of a certain cross of sweet peas should be red or white in a 3:1 ratio. That is, each plant has probability 3/4 of having red blossoms, and the blossom colors of separate plants are independent. (a) What is the probability that exactly three out of nine of these plants have red blossoms? (Round your answer to four decimal places.) (b) What is the mean number of red-blossomed plants when 64 plants of this type are grown from seeds? (Enter your answer to one decimal place.) (c) What is the probability of obtaining at least 48 red-blossomed plants when 64 plants are grown from seeds? Use the Normal approximation. (Round your answer to four decimal places.)

Answer 1

We will use the binomial law to solve the problem where P(X=x)= nCx* (p)^x * (q)^(n-x)

(a) here p= 3/4= 0.75, hence q= 1-0.75 = 0.25

Probability that exactly three out of nine plants have red blossoms= P(X=3)= 9C3* (0.75)^3 * (0.25)^6 = 0.0086 or 0.0087

(b) here n=64. Now mean for the binomial distribution is given by n*p. hence mean= 64*0.75= 48

(c) Using the normal approximation we have Z= (X- np)/ sqrt(npq).

now np= 64*0.75= 48 and npq= 64*0.75*0.25=12 and sqrt(npq) = sqrt(12) = 3.4641016151

hence Z= (48-48)/3.4641016151= 0

Now we need Prob(X>=48) = P(Z>=0) = 1 - P(Z<0) = 1- 0.5= 0.5