Bayes' theorem and partition of S

Question

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places. Y1, Y2, Y3 form a partition of S.

P(X | Y1) = .8, P(X | Y2) = .1, P(X | Y3) = .9, P(Y1) = .1, P(Y2) = .4.

Find P(Y1 | X).

P(Y1 | X) =

For this one I thought that all I had to do was P(X | Y1)*P(Y1)/P(X | Y1)*P(Y1)+P(X | Y2)*P(Y2)+P(X | Y3)*P(Y3)

But when I do that I am not getting the correct answer, is it possible that the value for P(Y3) is not .1 and if it is not, what is it?

Answer 1

When $Y_1,Y_2,Y_3$ form a partition of $S$, then their probabilities add up to one:

$P(Y_1)+P(Y_2)+P(Y_3)=1\Rightarrow P(Y_3)=1-0.1-0.4=0.5$

Answer 2

Actually I did the math incorrectly and P(Y3) is equal to .5 instead, with this new information the equation would work out correctly and the answer would come out to .1404