Question

There are 3 urns containing coins. Urn I contains 7 gold coins, urn II contains 1 gold coin and 1 silver coin, and urn III contains 6 silver coins. An urn is selected at random and a coin is drawn from the urn. If the selected coin is silver, what is the probability that urn III was selected? (Enter your probabilities as fractions.)

(a) Use a probability tree to find the probability.
(b) Use Bayes' formula to find the probability.

Answer 1

The probability that urn III was selected given that a silver coin is drawn is 2/3. This can be found using both a probability tree and Bayes' formula.

Step-by-step explanation:

(a) To find the probability that urn III was selected, we can use a probability tree. The first branch represents the probability of selecting each urn, which is 1/3 for each urn. The second branch represents the probability of selecting a silver coin from each urn. Since urn III only contains silver coins, the probability is 1. Therefore, the probability of selecting urn III given that a silver coin is drawn is:

P(III|S) = P(S|III) * P(III) / P(S) = (1 * 1/3) / (1/3 + 1/6) = 2/3.

(b) Alternatively, we can use Bayes' formula to find the probability. Bayes' formula states that P(A|B) = P(B|A) * P(A) / P(B), where A represents selecting urn III and B represents drawing a silver coin. Using the given information, we can substitute the values into the formula:

P(III|S) = P(S|III) * P(III) / P(S) = (1 * 1/3) / (1/3 + 1/6) = 2/3.