# Thread: Bayes' Theorem

1. ## Bayes' Theorem

Cheryl has a collection of 4 red dice and 5 blue dice. She picks one die at random and rolls it 34 times.
With a red die, the probability of rolling a four is 0.29. With a blue die, the probability of rolling a four is 0.26.
What is the probability that she used a red die if she gets a four on 6 of the 34 rolls?

I understand that I have to construct a tree diagram and use conditional probability but I seem to be getting the wrong answer each time I do it. Can anyone help me with steps?

2. ## Re: Bayes' Theorem

Originally Posted by ballerninja29
Cheryl has a collection of 4 red dice and 5 blue dice. She picks one die at random and rolls it 34 times.
With a red die, the probability of rolling a four is 0.29. With a blue die, the probability of rolling a four is 0.26.
What is the probability that she used a red die if she gets a four on 6 of the 34 rolls?
I understand that I have to construct a tree diagram and use conditional probability but I seem to be getting the wrong answer each time I do it.
Whoever set this question is a sadist. I don't think that a tree-diagram is the way to try.
You are looking for $\mathscr{P}(R|S)=\dfrac{\mathscr{P}(R\cap S)}{\mathscr{P}(S)}$ the probability that she used a red die if she gets a four on 6 of the 34 rolls.

$\mathscr{P}(S)=\mathscr{P}(S|R)\mathscr{P}(R)+ \mathscr{P}(S|B)\mathscr{P}(B)$

$\mathscr{P}(S|R)\mathscr{P}(R)= {34\choose 6}(0.29)^6(0.71)^{28}(0.5)$

Can you finish?