A marketing research shows that for every 100 flyers distributed

to potential customers, 1.2 customers will be captured. If 10000
flyers are being distributed, find
a. the probability that 150 customers are captured
b. the expected value of this probability distribution

i dont get the answer

what is the solution for the above question

To find the probability that 150 customers are captured, we need to use the Poisson distribution formula. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space when the average rate of occurrence is known and the events are independent.

The formula for the Poisson distribution is:

P(x; λ) = (e^(-λ) * λ^x) / x!

Where:
- P(x; λ) is the probability of x events occurring given an average rate of λ
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average rate of occurrence
- x is the number of events we're interested in

In this case, λ is the average number of customers captured per unit (100 flyers), which is given as 1.2. We can calculate the probability for 150 customers captured using the Poisson distribution formula.

a. Probability that 150 customers are captured:

P(150; 1.2) = (e^(-1.2) * 1.2^150) / 150!

To calculate this, you can substitute the values into the formula and solve it using a calculator or software that has the capability to evaluate mathematical functions.

b. The expected value of this probability distribution:

The expected value of a probability distribution is calculated by multiplying each possible outcome by its respective probability and summing them up. In this case, we need to calculate the expected number of customers captured.

Expected value = ∑(x * P(x; λ))

For a large number of flyers being distributed, we can approximate the Poisson distribution with a normal distribution using λ as the mean and λ as the variance. The expected value of a Poisson distribution is the mean itself.

Therefore, the expected value of the probability distribution in this case is 1.2 customers captured.