(a) If 10 auto insurance claims are taken at random, use the binomial distribution to determine the probability that:
(i) at least 6 claims are fraudulent
(ii) at most 3 claims are fraudulent
(iii) anywhere between 5 and 8 (inclusive) claims are fraudulent
(iv) at least 1 claim is fraudulent
To solve this problem using the binomial distribution, we need to know two things:
1. The number of trials (n): This is the number of auto insurance claims taken at random, which is given as 10 in the problem.
2. The probability of success (p): This is the probability that a single claim is fraudulent. Unfortunately, the problem does not provide this information. Without the probability, we cannot directly calculate the probabilities using the binomial distribution formula.
If the probability (p) for a single claim being fraudulent is given, we can proceed with the following steps:
Step 1: Identify the required probability mass function. In the binomial distribution, the probability mass function (P(x)) is given by:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where:
- C(n, x) is the number of combinations of n trials taken x at a time (which can be calculated as n! / (x! * (n-x)!))
- p^x is the probability of x successes (fraudulent claims)
- (1-p)^(n-x) is the probability of (n-x) failures (non-fraudulent claims)
Step 2: Solve for each probability using the given information and the formula above.
(i) At least 6 claims are fraudulent:
P(x >= 6) = P(x = 6) + P(x = 7) + P(x = 8) + P(x = 9) + P(x = 10)
(ii) At most 3 claims are fraudulent:
P(x <= 3) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)
(iii) Anywhere between 5 and 8 (inclusive) claims are fraudulent:
P(5 <= x <= 8) = P(x = 5) + P(x = 6) + P(x = 7) + P(x = 8)
(iv) At least 1 claim is fraudulent:
P(x >= 1) = 1 - P(x = 0)
Without the value of p, we cannot calculate the exact probabilities. You would need to obtain or estimate the probability of a single claim being fraudulent from the available data or context to proceed with solving this problem.