Financial Risk Manager Part 1
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There are three different bags. The first bag contains 3 square blocks and 2 round blocks. The second bag contains 2 square blocks and 3 round blocks. The third bag contains 5 round blocks. In an experiment, a bag is randomly chosen and then a block picked. Given a round block was selected, what is the probability it came from the second bag?
Explanation:
Explanation
This is a Bayes' Theorem problem where we need to find the conditional probability that the block came from the second bag given that a round block was selected.
Step 1: Define Events
- Let "1" be the event that the first bag is picked
- Let "2" be the event that the second bag is picked
- Let "3" be the event that the third bag is picked
- Let "R" be the event that a round block is drawn
Step 2: Apply Bayes' Theorem
We want: P(2|R) = P(R|2) × P(2) / P(R)
Step 3: Calculate Individual Probabilities
- P(1) = P(2) = P(3) = 1/3 (each bag equally likely to be chosen)
- P(R|1) = 2/5 (2 round blocks out of 5 total in bag 1)
- P(R|2) = 3/5 (3 round blocks out of 5 total in bag 2)
- P(R|3) = 5/5 = 1 (all blocks are round in bag 3)
Step 4: Calculate Total Probability P(R)
P(R) = P(R|1) × P(1) + P(R|2) × P(2) + P(R|3) × P(3) P(R) = (2/5) × (1/3) + (3/5) × (1/3) + 1 × (1/3) P(R) = 2/15 + 3/15 + 5/15 = 10/15 = 2/3
Step 5: Calculate Final Answer
P(2|R) = P(R|2) × P(2) / P(R) = (3/5) × (1/3) / (2/3) = (3/15) / (2/3) = (1/5) / (2/3) = (1/5) × (3/2) = 3/10
Therefore, the probability that the round block came from the second bag is 3/10.