Examples

Marble transferred, marble drawn

Setup:

• Bin 1 holds five red and four green marbles.
• Bin 2 holds four red and five green marbles.

Experiment:

• You take a random marble from Bin 1 and put it in Bin 2 and shake Bin 2.
• Then you draw a random marble from Bin 2 and look at it.

Questions:

(a) What is the probability you draw a red marble?

(b) Supposing that you drew a red marble, what is the probability that a red marble was transferred?

Solution

Construct the tree diagram:

Identify sub-experiments, label events, compute probabilities:

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(a) Compute $P[D_R]$:

Add up leaf numbers for $D_R$ at leaf:

$$P[D_R]=\frac{25}{90}+\frac{16}{90}=\frac{41}{90}$$

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(b) Compute $P[T_R|D_R]$:

Conditional probability definition:

$$\begin{array}{c}P[T_R|D_R]=\frac{P[T_R{D}_R]}{P[D_R]}\\ \\ \gg \gg \frac{25/90}{41/90}\gg \gg \frac{25}{41}\end{array}$$

Interpretation: value of desired path over value of all possible paths.