An auditor of Health Maintenance Services reports 40 percent of policyholders 55 years and older submit a claim during the year. Fifteen policyholders are selected at random for company records.
a)how many of the policyholders would you expect to have filed a claim within the last year?
b)what is the probability that 10 of the selected policyholders submitted a claim last year?
c)what is the probability that 10 or more of the selected policyholders filed a claim last year?
d)what is the probability that more than 10 of the selected policyholders filed a claim last year?
Find mean and standard deviation. Use z-scores to find z, then a z-table to find your probability based on the z-score. This is a normal approximation to the binomial distribution.
I'll get you started:
mean or expected value = np = (15)(.40) = ?
standard deviation = √npq = √(15)(.40)(.60) = ?
Note: q = 1 - p
Finish the calculations.
To find z-score:
z = (x - mean)/sd
Note: x = 10
Once you have the z-score, look in a z-table using the score to find your probability. Remember that the question is asking different questions when looking at the table.
Question , what does the letter q represent?
To solve these problems, we will use the binomial probability formula, which can be calculated using the formula:
P(x) = (nCr) * (p^x) * ((1-p)^(n-x))
Where:
P(x) is the probability of x successes,
n is the total number of trials,
p is the probability of success in each trial, and
nCr represents the combination formula for choosing x items from n items.
Given:
p = 0.40 (probability of policyholders 55 years and older submitting a claim)
n = 15 (number of policyholders selected randomly)
a) To calculate the expected number of policyholders who would have filed a claim within the last year, we simply multiply the probability (p) by the total number of policyholders (n):
Expected number = p * n = 0.40 * 15 = 6
Therefore, expected number of policyholders who filed a claim is 6.
b) To calculate the probability that exactly 10 policyholders submitted a claim last year, we substitute x = 10 into the formula:
P(10) = (15C10) * (0.40^10) * (0.60^5)
= (3003) * (0.40^10) * (0.60^5)
Using a calculator, compute (0.40^10) * (0.60^5) and multiply by 3003 to get the final probability value.
c) To calculate the probability that 10 or more of the selected policyholders filed a claim last year, we need to find the cumulative probability from x = 10 to x = 15. This can be calculated as:
P(10 or more) = P(10) + P(11) + ... + P(15)
Calculate each of these individual probabilities using the formula mentioned above and add them together.
d) To calculate the probability that more than 10 policyholders selected have filed a claim, we need to find the cumulative probability from x = 11 to x = 15. This can be calculated as:
P(more than 10) = P(11) + P(12) + ... + P(15)
Calculate each of these individual probabilities and add them together.
To answer these questions, we need to use the concept of probability. Probability can be calculated using a basic formula, which is the ratio of the number of favorable outcomes to the total number of possible outcomes.
In this case, we have been given that 40% of policyholders 55 years and older submit a claim during the year. Let's use this information to solve the questions.
a) To calculate the expected number of policyholders who filed a claim last year, we need to multiply the total number of policyholders by the probability of filing a claim. The total number of policyholders selected is 15, and the probability of filing a claim is 40%, which is 0.40. Therefore, the expected number of policyholders who filed a claim would be:
Expected number = Total number × Probability
Expected number = 15 × 0.40
Expected number = 6
So, we would expect that 6 policyholders out of the 15 selected would have filed a claim last year.
b) To calculate the probability that exactly 10 of the selected policyholders submitted a claim last year, we can use the binomial probability formula. The formula for calculating the probability of getting exactly x successes in n trials is:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
- n is the total number of trials (number of policyholders selected),
- x is the number of successes (policyholders who filed a claim),
- p is the probability of success (40% or 0.40),
- q is the probability of failure (60% or 0.60), which is the complement of p,
- (nCx) is the number of combinations of n items taken x at a time.
Applying this formula, we can calculate the probability of exactly 10 policyholders out of 15 submitting a claim last year.
P(10) = (15C10) * (0.40^10) * (0.60^(15-10))
P(10) = (15! / (10! * (15-10)!)) * (0.40^10) * (0.60^5)
You can use a calculator or software to evaluate this expression and find the numerical result.
c) To calculate the probability that 10 or more of the selected policyholders filed a claim last year, we need to calculate the probabilities for each number from 10 to 15 and then sum them up.
P(10 or more) = P(10) + P(11) + P(12) + P(13) + P(14) + P(15)
Calculate each P(x) as explained in question b) and sum them up to get the probability.
d) To calculate the probability that more than 10 of the selected policyholders filed a claim last year, we need to calculate the probabilities for numbers greater than 10 and then sum them up.
P(more than 10) = P(11) + P(12) + P(13) + P(14) + P(15)
Calculate each P(x) as explained in question b) and sum them up to get the probability.