https://file.aiquickdraw.com/mathbot/file/1728684510132bepvs2dw.jpg

Let's break down and solve the problem shown in the image.

### Given Information:
- A consumer organization provides the following estimates about car repairs in a year:
  - 15% of cars will need to be repaired once (probability = 0.15).
  - 7% will need repairs twice (probability = 0.07).
  - 2% will require three or more repairs (probability = 0.02).

### Complementary Probability:
The probability that a car does **not** need any repairs at all is the complement of the sum of the given probabilities:
\[
P(\text{no repair}) = 1 - (0.15 + 0.07 + 0.02) = 1 - 0.24 = 0.76
\]

Let's now solve the questions:

### (a) Probability that neither car will need repair:
If the probability that one car will not need repair is \( 0.76 \), the probability that neither of the two cars will need repair is:
\[
P(\text{neither}) = 0.76 \times 0.76 = 0.5776
\]

### (b) Probability that both cars will need repair:
The probability that one car will need repair is \( 1 - 0.76 = 0.24 \). The probability that both cars will need repair is:
\[
P(\text{both}) = 0.24 \times 0.24 = 0.0576
\]

### (c) Probability that at least one car will need repair:
This is the complement of the probability that neither car will need repair:
\[
P(\text{at least one}) = 1 - P(\text{neither}) = 1 - 0.5776 = 0.4224
\]

### Final answers:
- (a) \( P(\text{neither}) = 0.5776 \)
- (b) \( P(\text{both}) = 0.0576 \)
- (c) \( P(\text{at least one}) = 0.4224 \)

Would you like more details or further clarifications on any of these steps?

Here are 5 related questions to extend this concept:
1. What is the probability that exactly one of the cars will need repair?
2. If you own three cars, what is the probability that none of them will need repair?
3. How does the probability change if the likelihood of needing repair increases?
4. What is the expected number of cars needing repair if you own two cars?
5. Can you model this scenario with more than two cars using binomial distribution?

**Tip:** Always look for complementary probabilities when asked for "at least one" scenarios.

A consumer organization estimates that over a 1-year period 15% of cars will need to be repaired once, 7% will need repairs twice, and 2% will require three or more repairs. If you own two cars, what is the probability that (a) neither will need repair? (b) both will need repair? (c) at least one car will need repair?

Learn how to calculate the probability of needing repairs for two cars based on repair rates. This problem involves complementary probability concepts and includes key formulas for finding the chance of neither, both, or at least one car needing repair.

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