Question

Genetic theory predicts that, in the second generation of a cross of sweet pea plants, flowers will be either red or white, with each plant having a 25% chance of producing red flowers. Flower colours of separate plants are independent. Let X be the number of plants with red flowers out of 20 plants selected at random from the second generation of this cross. (a) What is the probability distribution of X? [3] (b) Calculate: (i) the mean and standard deviation of X. [3] (ii) P( X > 8) [2] (iii) P( 4 ≤ X ≤ 10) [2] (c) If only 3 of the 20 plants had red flowers, would this be an unusual sample? Calculate a probability and use it to justify your answer

Answer 1

(a)

Probability Distribution of X = the number of plants with red flowers out of 20 plants selected at random from the second generation of this cross is Binomial Distribution with n = Number of trials = 20, p = Probability of success in a single trial (Probability of red flowers) = 0.25 and q = Probability of failure in a single trial (Probability of white flowers) = 0.75.

(b)

(i) Mean = = np = 20 X 0.25 = 5

standard deviation of X =

(ii)

P(X>8) = 1 - [P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)+P(X=8)]

So,

P(X>8) = 1 - 0.9591

= 0.0409

So,

Answer is:

0.0409

(iii) P(4X10) = P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

So,

P(4X10) = 0.7709

So,

Answer is:

0.7709

(c)

= 13.39 % > 5 %

Since the probability is greater than 5%, this would not be an unusual sample