Only 0.02% of credit card holders of a company report the loss or theft of their credit cards each month. The company has 15,000 credit cards in the city of Memphis. Use the Poisson probability tables to answer the following questions. What is the probability that during the next month in the city of Memphis

a. no one reports the loss or theft of their credit cards?
b. every credit card is lost or stolen?
c. six people report the loss or theft of their credit cards?
d. at least nine people report the loss or theft of their cards?
e. Determine the expected number of reported lost or stolen credit cards.
f. Determine the standard deviation of the number of reported lost or stolen cards.

a. 0.9998

b. 0.0002
c. 0.0504
d. 0.0224
e. 3 cards
f. 1.731877594 (rounded 1.7319)

I apologize, 3 of the above answers are not correct.

* a. 0.0498
* b. 0.0000
c. correct
* d. 0.0038
e. correct
f. correct

Only 0.02% of credit card holders of a company report the loss or theft of their credit cards each month. The company has 15,000 credit cards in the city of Memphis. What is the probability that during the next month in the city of Memphis

a. The probability that no one reports the loss or theft of their credit cards can be calculated using the Poisson distribution formula:

P(x = 0) = (e^(-λ) * λ^x) / x!

where λ is the average number of events per time period, which is equal to the product of the total number of credit cards (15,000) and the reported loss or theft rate (0.02% or 0.0002).

P(x = 0) = (e^(-λ) * λ^0) / 0!
= e^(-λ)

Substituting the values:

P(x = 0) = e^(-0.0002 * 15,000)

b. The probability that every credit card is lost or stolen can also be calculated using the Poisson distribution formula:

P(x = 15,000) = (e^(-λ) * λ^x) / x!

Substituting the values:

P(x = 15,000) = (e^(-0.0002 * 15,000) * (0.0002 * 15,000)^15,000) / 15,000!

c. The probability that exactly six people report the loss or theft of their credit cards can be calculated using the Poisson distribution formula:

P(x = 6) = (e^(-λ) * λ^x) / x!

Substituting the values:

P(x = 6) = (e^(-0.0002 * 15,000) * (0.0002 * 15,000)^6) / 6!

d. The probability that at least nine people report the loss or theft of their credit cards can be calculated by summing the probabilities of exactly nine, ten, eleven, and so on, up to the maximum number of credit card holders (15,000), using the Poisson distribution formula for each value, and then subtracting the sum from 1:

P(x >= 9) = 1 - (P(x = 0) + P(x = 1) + P(x = 2) + ... + P(x = 8))

e. The expected number of reported lost or stolen credit cards can be calculated as the mean of the Poisson distribution:

E(x) = λ

Substituting the value:

E(x) = 0.0002 * 15,000

f. The standard deviation of the number of reported lost or stolen credit cards can be calculated as the square root of the variance of the Poisson distribution:

Standard Deviation = sqrt(λ)

Substituting the value:

Standard Deviation = sqrt(0.0002 * 15,000)

To answer these questions using the Poisson probability tables, we need to calculate the probabilities using the given information.

First, let's define the parameters of our Poisson distribution:
λ = average number of credit cards reported lost or stolen per month = 0.02% of 15,000 = 0.02/100 * 15000 = 3

a. To find the probability that no one reports the loss or theft of their credit cards, we can use the Poisson probability formula:
P(X = 0) = (e^(-λ)) * (λ^0) / 0!

Substituting λ = 3 into the formula:
P(X = 0) = (e^(-3)) * (3^0) / 0! = e^(-3) ≈ 0.0498

So, the probability that no one reports the loss or theft of their credit cards is approximately 0.0498.

b. To find the probability that every credit card is lost or stolen, we can use the Poisson probability formula:
P(X = 15000) = (e^(-λ)) * (λ^15000) / 15000!

Substituting λ = 3 into the formula:
P(X = 15000) = (e^(-3)) * (3^15000) / 15000!

Since calculating this directly would be challenging, we can approximate it to 0 because the chance of such an event happening is exceptionally low.

c. To find the probability that six people report the loss or theft of their credit cards, we can again use the Poisson probability formula:
P(X = 6) = (e^(-λ)) * (λ^6) / 6!

Substituting λ = 3 into the formula:
P(X = 6) = (e^(-3)) * (3^6) / 6!

Calculating this out, we find that P(X = 6) is approximately 0.0504.

d. To find the probability that at least nine people report the loss or theft of their credit cards, we need to sum up the probabilities of all the events where X is greater than or equal to nine:
P(X ≥ 9) = 1 - P(X < 9)

We can calculate P(X < 9) by summing the probabilities of X = 0, 1, 2, ..., 8.

P(X < 9) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 8)

Using the Poisson probability formula for each of these values, we can calculate this sum.

e. The expected number of reported lost or stolen credit cards can be calculated using the formula for the mean of a Poisson distribution:
Mean (μ) = λ

Since λ = 3, the expected number of reported lost or stolen credit cards is also 3.

f. The standard deviation of the number of reported lost or stolen cards can be calculated using the formula for the standard deviation of a Poisson distribution:
Standard Deviation (σ) = √μ

Since μ = 3, the standard deviation of the number of reported lost or stolen cards is √3 ≈ 1.732.