Explanation

This problem involves calculating the probability that no one will have a claim using conditional probability and the law of total probability.

Given Information:

• P(Male) = 55% = 0.55
• P(Female) = 45% = 0.45
• P(Claim|Male) = 10% = 0.10
• P(Claim|Female) = 7% = 0.07

Step 1: Calculate Probability of Claim for Each Gender

Using the conditional probability formula:

$P(\text{Claim} \cap \text{Male}) = P(\text{Claim}|\text{Male}) \times P(\text{Male}) = 0.10 \times 0.55 = 0.055$

$P(\text{Claim} \cap \text{Female}) = P(\text{Claim}|\text{Female}) \times P(\text{Female}) = 0.07 \times 0.45 = 0.0315$

Step 2: Calculate Total Probability of Claim

Using the law of total probability:

$P(\text{Claim}) = P(\text{Claim} \cap \text{Male}) + P(\text{Claim} \cap \text{Female}) = 0.055 + 0.0315 = 0.0865$

Step 3: Calculate Probability of No Claim

Since the probability of no claim is the complement of having a claim:

$P(\text{No Claim}) = 1 - P(\text{Claim}) = 1 - 0.0865 = 0.9135 = 91.35\%$

Rounded to the nearest whole percentage, this gives us 91%.

Verification:

• Male no claim probability: 0.55 × (1 - 0.10) = 0.55 × 0.90 = 0.495
• Female no claim probability: 0.45 × (1 - 0.07) = 0.45 × 0.93 = 0.4185
• Total no claim probability: 0.495 + 0.4185 = 0.9135 = 91.35%

Both methods confirm the answer is 91%.