In planning a family of 4 children assume boys and girls are equally likely on each birth. Find (i) the probability distribution (ii) the expected number of boys (iii) the standard deviation of the number of boys.

To find the probability distribution, we need to calculate the probabilities of each possible outcome - that is, the number of boys, ranging from 0 to 4.

(i) Probability Distribution:
Let's label the number of boys from 0 to 4 as B and calculate the probabilities of each case:
P(B = 0) = P(girl, girl, girl, girl) = (1/2) * (1/2) * (1/2) * (1/2) = 1/16
P(B = 1) = P(boy, girl, girl, girl) + P(girl, boy, girl, girl) + P(girl, girl, boy, girl) + P(girl, girl, girl, boy) = 4 * (1/2) * (1/2) * (1/2) * (1/2) = 4/16
P(B = 2) = P(boy, boy, girl, girl) + P(boy, girl, boy, girl) + P(boy, girl, girl, boy) + P(girl, boy, boy, girl) + P(girl, boy, girl, boy) + P(girl, girl, boy, boy) = 6 * (1/2) * (1/2) * (1/2) * (1/2) = 6/16
P(B = 3) = P(boy, boy, boy, girl) + P(boy, boy, girl, boy) + P(boy, girl, boy, boy) + P(girl, boy, boy, boy) = 4 * (1/2) * (1/2) * (1/2) * (1/2) = 4/16
P(B = 4) = P(boy, boy, boy, boy) = (1/2) * (1/2) * (1/2) * (1/2) = 1/16

So, the probability distribution is:
B = 0: P(B = 0) = 1/16
B = 1: P(B = 1) = 4/16
B = 2: P(B = 2) = 6/16
B = 3: P(B = 3) = 4/16
B = 4: P(B = 4) = 1/16

(ii) Expected number of boys:
The expected number of boys (E[B]) is calculated by multiplying each possible value of boys by its corresponding probability and summing them up.

E[B] = (0 * P(B = 0)) + (1 * P(B = 1)) + (2 * P(B = 2)) + (3 * P(B = 3)) + (4 * P(B = 4))
= (0 * 1/16) + (1 * 4/16) + (2 * 6/16) + (3 * 4/16) + (4 * 1/16)
= 0 + 4/16 + 12/16 + 12/16 + 4/16
= 32/16
= 2

Therefore, the expected number of boys is 2.

(iii) Standard deviation of the number of boys:
To find the standard deviation, we need to calculate the variance first.

Variance = Σ (x - μ)^2 * P(x)
where x denotes each possible value (0, 1, 2, 3, 4), μ is the expected number of boys (2), and P(x) is the probability of each corresponding value.

Variance = (0 - 2)^2 * (1/16) + (1 - 2)^2 * (4/16) + (2 - 2)^2 * (6/16) + (3 - 2)^2 * (4/16) + (4 - 2)^2 * (1/16)
= (-2)^2 * (1/16) + (-1)^2 * (4/16) + (0)^2 * (6/16) + (1)^2 * (4/16) + (2)^2 * (1/16)
= 4/16 + 4/16 + 0 + 4/16 + 4/16
= 16/16
= 1

The standard deviation (σ) is the square root of the variance.

σ = √1
= 1

Therefore, the standard deviation of the number of boys is 1.