Assume a family is planning to have three children.

1. Why do probabilities centered around this scenario represent the same probabilities as those for flipping three coins

mmmh,

in flipping one coin, and having one child:
prob(heads) = 1/2, prob(tails) = 1/2
prob(boy) = 1/2 , prob (girl) = 1/2

for flipping 3 coins, or considering 3 kids:
prob(1 head, 2 tails) = 3(1/2)^3 = 3/8
prob(1 boy, 2 girls) = 3(1/2)^3 = 3/8

etc.

This is because each pregnancy is independent of the others so the chances of receiving either a boy or girl is 50% for each separate pregnancy. the same as flipping three coins, the chances of having either heads or tails is 50% and each flip is independent of the others.

The scenario of a family planning to have three children can be thought of as similar to flipping three coins because both involve a series of independent events, each with two possible outcomes.

When flipping a fair coin, there are two possible outcomes: heads (H) or tails (T). Similarly, when a family plans to have children, there are two possible outcomes for each child: a boy (B) or a girl (G).

Now, to calculate the probabilities, we can use the concept of sample space. In the case of flipping three coins, the sample space consists of all possible outcomes: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Each outcome represents the potential combination of heads (H) and tails (T) after three coin flips.

Similarly, in the case of having three children, the sample space consists of all possible outcomes: {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}. Each outcome represents the potential combination of boys (B) and girls (G) in a family of three children.

By comparing both sample spaces, we can see that they are identical, meaning that the number of outcomes and their probabilities are the same for both scenarios. Thus, the probabilities centered around the family scenario represent the same probabilities as those for flipping three coins.