Bayesian urn questions

Question

There are two urns, each with four ping-pong balls. In one urn, three of the balls are red, and one is white; in the other, three are white, and one is red. Without knowing which urn you are choosing, you reach inside and draw out, at random, a red ball. You put the ball back into the urn, mix up the contents, and repeat the experiment (remember there are four balls once more inside the urn). Again you draw out a red ball. Based on this two-part experiment, what is the probability that the urn you chose is the one with the three red balls in it?

I'm not entirely too sure how to start this. If anyone can point me in the right direction, I'd really appreciate it.

Answer 1

Just hints, as requested:

Let $U_i$ be the event that you are drawing from urn $i$ and let $R_i$ be the event that you draw a red ball on your ith draw (with replacement).

Here are some questions to guide your thinking:

1. If you know you are drawing from a particular urn, with replacement, what is the applicable probability model for drawing red balls?
2. Lets assume $P(U_i)=0.5$, what is the probability of drawing two red balls, overall? (Hint: this is the denominator of Bayes theorem). Think about the law of total probability: Your events can be partitioned into "draws from $U_1$" and "draws from $U_20$"

Let me know if you need more, but the above should get you to the solution.

Answer 2

By using Bayes' Theorem, what I am getting is that $$P(\mathbf{Urn1} | RR) = \frac{(P(RR|\mathbf{Urn1})) * P(\mathbf{Urn1})}{P(RR)}$$ Where $P(RR|\mathbf{Urn1}) = 9/16 $ and $P(\mathbf{Urn1}) = 1/2$.

To find $P(RR)$ we use the law of total probability: $$P(RR) =P(RR|\mathbf{Urn1}) *P(\mathbf{Urn1}) + P(RR|\mathbf{Urn2}) * P(\mathbf{Urn2})$$

We know that $P(RR|\mathbf{Urn2}) = 1/16$ and $P(\mathbf{Urn2})$ is also $1/2$

Therefore we get that $$P(RR) = (9/16)(1/2) + (1/16)(1/2) = .3125 $$

Plug this back into the final equation to get $$P(\mathbf{Urn1} | RR) = \frac{(9/16)* (1/2)}{.3125} = 0.9$$

There is a 90% chance that you picked the urn with 3 red balls in it.