Consider the following event for a family with children

Question

Consider the following event for a family with children: A={children of both sexes} , B={at most one boy}. a. show that A and B are independent events if a family has three children. b. show that A and B are dependent events if a family has only two children.

Answer 1

Two children: $S_2 = \{FF, FM, MF, MM\},$ elements equally likely. $P(A) = P(\{FM, MF\}) = 2/4 = 1/2;$ $P(B) = P(\{FF, FM, MF\}) = 3/8;$ and $P(A\cap B) = P(\{FM, MF\}) = 2/4 = 1/2 \ne P(A)P(B) = 3/16.$ So $A$ and $B$ not independent.

Three children: $S_3 = \{FFF, FFM, FMF, MFF, FMM, FMF, MMF, MMM\}.$ I got $P(A) = 3/4,\, P(B) = 1/2,\, P(A\cap B) = 3/8.$ So $A$ and $B$ are independent. I will leave the counting and multiplying to you. Enjoy!

Addendum: Simulation of a million families of both sizes gives probabilities in close agreement with results above. (It is reasonable to expect two or three place accuracy.)

set.seed(404);  m = 10^6;  n = 2
x = replicate(m, sum(sample(0:1, n, rep=T)) ) # nr boys
A = (x > 0) & (x < n);  B = (x <= 1);  AB = A&B 
p.A = mean(A);  p.B = mean(B);  p.A*p.B;  p.AB = mean(AB)
## 0.3747197
p.AB
## 0.499611  # Disagrees with above; not A and B not independent
p.A;  p.B
## 0.499611
## 0.750023

n = 3
y = replicate(m, sum(sample(0:1, n, rep=T)) ) # nr boys
A = (y > 0) & (y < n);  B = (y <= 1);  AB = A&B 
p.A = mean(A);  p.B = mean(B);  p.A*p.B;  p.AB = mean(AB)
## 0.3749833
p.AB;  3/8
## 0.374658 # agrees with above (within marg of sim err);  suggests indep
## 0.375
p.A;  p.B
## 0.749436
## 0.500354