Question

The Geometric Distribution The Geometric Distribution is a Discrete Probability Distribution that is commonly applied when a series of trials/experiments will produce a "success" or a "failure" (a binary outcome). Thie Geometric Distribution is different from the Binomial Distribution in that it is concerned with how many failures occur before an initial success.

This type of distribution may be used if the following conditions apply:
- Each observation is independent.
- Each observation has only one of two outcomes ("success" or "failure").
- The probability of a "success" (p) is the same for each trial/experiment/observation/outcome.
Apply the Geometric Distribution to a scenario.
Suppose a regional poll reveals that 75% of small business advertise in the local newspaper. Suppose a list of local business are called until one is found that advertises in the local newspaper.
The conditions of a Geometric Distribution are met: -
The decision of one small business to advertise doesn't affect the others. (independent observations)
- Each business does or does not advertise in the local paper. (binary outcomes)
- The probabitity of a "success" is the same for each observation. (p=0.75)
a. What is the probability that the third business called advertises in the local newspaper?
P 1 =
b. What is the probability that the business is found on the first, second, or third call?
P 2=
c. What is the probability that the business is found in more than 4 calls?
P 3=
d. What is the probability that the business is found between 3 and 9 calls?
P 4=

Answer 1

The Geometric Distribution is a discrete probability distribution that is concerned with the number of failures before an initial success. In this scenario, the probability of success is 0.75. We can use the geometric distribution formula to find probabilities for different scenarios.

Step-by-step explanation:

The Geometric Distribution is a discrete probability distribution that is commonly applied when a series of trials/experiments will produce a "success" or a "failure" (a binary outcome). It is concerned with how many failures occur before an initial success. In this scenario, the probability of success (p) is 0.75.

a. To find the probability that the third business advertises in the local newspaper, we use the formula P(X = k) = (1-p)^(k-1) * p, where X represents the number of trials until the first success. In this case, k = 3. Plugging in the values, we get: P(X = 3) = (1-0.75)^(3-1) * 0.75 = 0.046875.

b. To find the probability that the business is found on the first, second, or third call, we can add the probabilities of each event. P(X <= 3) = P(X = 1) + P(X = 2) + P(X = 3) = p + (1-p)*p + (1-p)^2*p

c. To find the probability that the business is found in more than 4 calls, we can calculate P(X > 4) = 1 - P(X <= 4).

d. To find the probability that the business is found between 3 and 9 calls, we can calculate P(3 <= X <= 9) = P(X <= 9) - P(X <= 2).