The Father and the Coins

These are not trick questions.

Your father tells you that long ago, he had two partners named Heather and Talia. He wanted to have children but couldn’t decide with whom; he resolved to leave it up to chance. He decided to flip his lucky coin (which is a fair coin), with the intention to:

have children with both women if the coin came up heads, or
have children with only one if the coin came up tails, flipping the coin again to decide with whom (heads for Heather, tails for Talia).

He tells you that in fact you were born as a result of this.

Question 1

What is the chance that all those years ago, the first coin flip came up Heads?

Answer: Of course, the chance of a fair coin landing heads is 50%.

Question 2

Suppose that you know that your mother is Heather. What is the chance that you have a half-sibling?

Answer: The coin flips were either Heads (50%, in which case you have a half-sibling), or Tails-Heads (25%, in which case you do not). Tails-Tails is impossible, as you would not have been born. The answer is therefore two in three.

Question 3

Suppose that you know that your mother is Talia. What is the chance that you have a half-sibling?

Answer: There was nothing special about either possible mother: the chance is still two in three, by the exact same logic as in the previous question.

Question 4

Suppose you were abandoned at birth and do not know the name of your mother. What is the chance that you have a half-sibling?

Answer: The chance is still two in three: it does not change regardless of who your mother is, so finding out cannot possibly change your belief in the probability.

Question 5

Relate the outcome of the coin flips to you having or not having a half-sibling. Under what circumstances do you have a half-sibling?

Answer: You have a half-sibling if and only if your father had a child with both women: that is, if the first coin came up Heads.

Question 6

What is the chance that all those years ago, the first coin flip came up Heads?

Answer: Of course, this is the chance of you having had a half-sibling, which we calculated in question 4 to be two in three.