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Difficulty: HardConditional Probability

An analyst at an investment firm evaluates a portfolio of 160160 corporate bonds for two potential risk factors: credit rating downgrade risk and liquidity risk. The evaluation reveals that 6060 bonds have credit rating downgrade risk, 7272 bonds have liquidity risk, and 6464 bonds have neither risk factor. If a bond is selected at random from those in the portfolio that have at least one of the two risk factors, what is the probability that it has credit rating downgrade risk? Express your answer as a decimal.

Answer: 0.625

Answer

The probability is 0.625 (or 5/8).
To calculate the probability that a bond has credit rating downgrade risk given that it has at least one risk factor, the sample space must be restricted to bonds with at least one risk factor. Out of 160160 bonds, 6464 have neither risk factor, leaving 160−64=96160 - 64 = 96 bonds with at least one risk factor. All 6060 bonds with credit rating downgrade risk are part of this group. The required conditional probability is 6096=58=0.625\frac{60}{96} = \frac{5}{8} = 0.625.

Step-by-Step Solution

1
Calculate the size of the restricted sample space (bonds with at least one risk factor).
Total bonds with at least one risk factor = 160 - 64 = 96 bonds.
The condition specifies that the selection is made only from bonds having at least one risk factor.
2
Identify the number of favorable outcomes within this restricted sample space.
Number of bonds with credit rating downgrade risk = 60.
All 60 bonds with credit rating downgrade risk inherently possess at least one risk factor, so they lie entirely within the restricted sample space.
3
Compute the conditional probability P(Downgrade Risk | At Least One Risk).
60 / 96 = 5 / 8 = 0.625.
Conditional probability requires dividing the count of favorable outcomes by the count of the restricted sample space.

Key Concept

Conditional Probability and Sample Space Restriction
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Topics

Conditional Probability

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