# The Pixie and the Sprite

by Jason Ross

The solution to this problem is found through conditional probability.
To begin with, there are 11 possibilities for the number of times out of 10 the pixie lies. For each of these possibilities, find the probability that the pixie would be able to make the statement "I lie exactly 4 out of 10 times." For example, in the case that the pixie actually lies 7 of 10 times, there is a 7/10 probability that he is able to make the (false) statement "I lie exactly 4 out of 10 times." The probability of the pixie being able to make the statement is the same as the probability of the pixie lying, except for the case that he is telling the truth (4/10); in that case, the probability of him being able to make that statement is 6/10 (which is equal to the probability of him being able to make a true statement).
Now, conditional probability says that the probability of the pixie lying X number of times out of 10 in his life is equal to the probability of him being able to make the statement (X/10, unless X is 4) divided by the sum of probabilities for all the cases. So, for our example, the probability that the pixie tells lies 7 of 10 times is:
(7/10) / (0/10 + 1/10 + 2/10 + 3/10 + 6/10 + 5/10 + 6/10 + 7/10 + 8/10 + 9/10 + 10/10) = 7/57
Similarly, the probability that the pixie lies X of 10 times is:
X/57, for X not equal to 4,
6/57, for X equal to 4.
Now we can find the probability of the pixie lying in his first statement. There is a X/57 chance (or 6/57 for the 4 of 10 case) that the pixie lies X of 10 times, and for each of these cases there is a X/10 chance that the pixie lied about the path. To find the odds that the pixie lied, simply add the products of probabilities for each case:
0/570 + 1/570 + 4/570 + 9/570 + 24/570 + 25/570 + 36/570 + 49/570 + 64/570 + 81/570 + 100/570 = 393/570 = 68.947%.
So, the probability the pixie lied is 68.947%, and you should go right.