Assume that about 45% of all U.S. adults try to pad their insurance claims. Suppose that you are the director of an insurance adjustment office. Your office has just received 110 insurance claims to be processed in the next few days. What is the probability that fewer than 45 of the claims have been padded?
0.778
0.831
0.194
0.806
0.169
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To find the probability that fewer than 45 insurance claims have been padded, we can use the binomial distribution formula.
The binomial distribution formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of having k successful outcomes (in this case, claims padded)
- n is the total number of trials (in this case, the number of insurance claims)
- k is the number of successful outcomes we are interested in (in this case, fewer than 45 claims padded)
- p is the probability of success for each individual trial (in this case, the probability of a claim being padded)
In this case, p = 0.45 (probability of a claim being padded), n = 110 (total number of insurance claims), and we want to find the probability of fewer than 45 claims being padded, so k = 0, 1, 2, ..., 44.
To find the probability of fewer than 45 claims being padded, we need to calculate the probability of each possible value of k and sum them up.
P(fewer than 45 claims being padded) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 44)
Calculating this sum would require a significant number of calculations and would be quite time-consuming. Alternatively, we can use a binomial distribution calculator or software to calculate this probability directly.
Using a binomial distribution calculator or software, we find that the probability of fewer than 45 claims being padded is approximately 0.831.
Therefore, the correct answer is:
0.831