SOLUTION: An analyst estimates that the probability of default on a seven-year AA-rated bond is 0.47, while that on a seven-year A-rated bond is 0.53. The probability that they will both def

Question 1162634: An analyst estimates that the probability of default on a seven-year AA-rated bond is 0.47, while that on a seven-year A-rated bond is 0.53. The probability that they will both default is 0.45.
a. What is the probability that at least one of the bonds defaults? (Round your answer to 2 decimal places.)

b. What is the probability that neither the seven-year AA-rated bond nor the seven-year A-rated bond defaults? (Round your answer to 2 decimal places.)

c. Given that the seven-year AA-rated bond defaults, what is the probability that the seven-year A-rated bond also defaults? (Round your answer to 2 decimal places.)

Answer by CPhill(2264) (Show Source): You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
p_aa = 0.47
p_a = 0.53
p_both = 0.45
# a. At least one defaults
p_at_least_one = p_aa + p_a - p_both
# b. Neither defaults
p_neither = 1 - p_at_least_one
# c. P(A defaults | AA defaults)
p_a_given_aa = p_both / p_aa
print(f"{p_at_least_one=}")
print(f"{p_neither=}")
print(f"{p_a_given_aa=}")
```
```text?code_stdout&code_event_index=2
p_at_least_one=0.55
p_neither=0.44999999999999996
p_a_given_aa=0.9574468085106383
```
To find the requested probabilities, we define the following events:
* $P(AA) = 0.47$: Probability that the seven-year AA-rated bond defaults.
* $P(A) = 0.53$: Probability that the seven-year A-rated bond defaults.
* $P(AA \cap A) = 0.45$: Probability that both bonds default.
### a. Probability that at least one of the bonds defaults
The probability that at least one bond defaults is represented by the union of the two events, $P(AA \cup A)$. We use the Addition Rule for Probability:
$$P(AA \cup A) = P(AA) + P(A) - P(AA \cap A)$$
$$P(AA \cup A) = 0.47 + 0.53 - 0.45$$
$$P(AA \cup A) = 1.00 - 0.45 = 0.55$$
**Probability that at least one defaults: 0.55**
---
### b. Probability that neither bond defaults
The event that neither bond defaults is the complement of the event that at least one defaults.
$$P(\text{neither}) = 1 - P(AA \cup A)$$
$$P(\text{neither}) = 1 - 0.55 = 0.45$$
**Probability that neither defaults: 0.45**
---
### c. Probability that the A-rated bond defaults given the AA-rated bond defaults
This is a conditional probability, denoted as $P(A | AA)$. The formula for conditional probability is:
$$P(A | AA) = \frac{P(A \cap AA)}{P(AA)}$$
$$P(A | AA) = \frac{0.45}{0.47} \approx 0.9574$$
Rounding to 2 decimal places:
**Probability: 0.96**
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