# Tossing coins, one fair, one unfair: how to represent the probability tree?

Suppose I have a fair coin and an unfair coin. The fair coin has head, tail, the unfair coin has both heads.

You pick one coin at random and toss them two times and observe the outcomes.

Which of the figure below is a better probability tree representation of these experiments (top or bottom)?

I am raising the question because, in the first figure, it seems to draw the head twice is redundant, given that the probability of getting a head is certainty.

• I think if you wanted to make it so each branch of the tree was a “possibility”, it might make a little more sense to put them as ordered pairs of the form $(P(FAIR=x), P(UNFAIR=x))$, as to avoid the redundancy of the “fair” and “unfair” branches at the very beginning. Otherwise as you said, the 2nd one is fine. – DaveNine Sep 30 '17 at 18:27

You have to remember that the total possibilities include 4 different sides: the fair coin's heads, the fair coin's tails, the unfair coin's head side A, and the unfair coin's head side B.

When calculating probabilities, you must, of course, consider total possibilities, and that includes the two possible different heads. Just because you can't distinguish between them, doesn't mean they aren't relevant.

Your first tree is the way to go for working out probabilities correctly.

Art of the Problem's Bayes Theorem video explains this exact problem in more detail. For the more general problem of why you should take different sides into account, even when they're the same, read Martin Gardner's "Three Card Swindle" (pp 93-95).

I think you interpreted it wrong for unfair coin probablity of getting head is greater than or less than half not equal to half also it may not be equal to one so your second tree emphasize on only one condition that coin is unfair with all head with probablity 1 which not seems to look correct.This is my thought maybe you are also right.

Unless you can and want to distinguish between the two different heads on the unfair coin, the second one is just fine, and of course a lot easier to use.

A question though: your tree represents that first you randomly pick one of the coins, after which you toss that coin two times ... is that what was meant?