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Let there be two buckets. Bucket I contains balls of 4 white, 1 red. Bucket II contains balls of 3 white, 5 red and 1 black. Consider the following procedure:

  1. You draw a ball uniformly at random from Bucket I, put it in Bucket II.
  2. Then, you draw a ball uniformly at random from Bucket II, put it in Bucket I. The random draw in step 2 is independent of the random draw in step 1.

(a) What is the probability that the resulting Bucket I contains balls of all 3 colours?

(b) What is the probability that the resulting Bucket I contains balls of only 1 colour?

(c) What is the probability that the resulting Bucket I still contains 4 white, and 1 red?

^ For the above questions, I am not sure if I am thinking too much by using conditional probability or if can I just draw out a probability tree to visualise the outcomes.

jbb
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1 Answers1

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Let $U$ be the colour of the ball drawn from bucket I, and $V$ be the colour drawn from bucket II, with values in $\{w, r, b\}$ (white, red, and black).

^ For the above questions, I am not sure if I am thinking too much by using conditional probability or if can I just draw out a probability tree to visualise the outcomes.

These are just the same task; since the probabilities you assign to branches on the tree are conditional probabilities, and the probabilities you calculate for the leaves are the resultant joint probability

$$\begin{array}{|l:l|l|}\hline\mathsf P(U=w)=4/5&\mathsf P(V=w\mid U=w)=\boxed?&\mathsf P(U=w\cap V=w)=\boxed?\\&\mathsf P(V=r\,\mid U=w)=\boxed?&\mathsf P(U=w\cap V=r)\,=\boxed?\\&\mathsf P(V=b\,\mid U=w)=\boxed?&\mathsf P(U=w\cap V=b)\,=\boxed?\\\hline\mathsf P(U=r)=1/5&\mathsf P(V=w\mid U=r)=\boxed?&\mathsf P(U=r\,\cap V=w)=\boxed?\\&\mathsf P(V=r\,\mid U=r)=\boxed?&\mathsf P(U=r\,\cap V=w)=\boxed?\\&\mathsf P(V=b\,\mid U=r)=\boxed?&\mathsf P(U=r\,\cap V=b)\,=\boxed?\\\hline\end{array}$$

You move a white, or red, ball from bucket I into bucket II, then move a ball back into bucket I. The probabilities of the colour moved outof bucket II will be conditional on the colour moved into it (thus these steps are not independent draws).

Graham Kemp
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