SOLUTION: 1. Assume that a procedure yields a binomial distribution with a trial repeated n=18 times. Use either the binomial probability formula (or technology) to find the probability o

Question 1191339: 1. Assume that a procedure yields a binomial distribution with a trial repeated
n=18 times. Use either the binomial probability formula (or technology) to find the probability of k=4 successes given the probability p=0.41 of success on a single trial.
(Report answer accurate to 4 decimal places.)
2. Assume that a procedure yields a binomial distribution with a trial repeated
n=11times. Use the binomial probability formula to find the probability of k=7 successes given the probability p= 2/3 of success on a single trial.
(Report answer accurate to 4 decimal places.)
3. Suppose that a box contains 7 cameras and that 4 of them are defective. A sample of 2 cameras is selected at random with replacement. Define the random variable X as the number of defective cameras in the sample.
a. Write the binomial probability distribution for X. Round to two decimal places.
b. What is the expected value of X? Round to two decimal places.

4. An insurance company prices its Tornado Insurance using the following assumptions:
In any calendar year, there can be at most one tornado.
In any calendar year, the probability of a tornado is 0.11.
The number of tornadoes in any calendar year is independent of the number of tornados in any other calendar year.
Using the insurance company's assumptions, calculate the probability that there are fewer than 4 tornadoes in a 23-year period. Round your answer to four decimal places.

5. A salesperson makes 7 sales per day on average. Use the formula for Poisson probabilities
P
(
k
)
=
λ
k
e
−
λ
k
!

to find the probability that the salesperson makes exactly 6 sales on a given day.
(a) First fill in the details of the binomial probability formula:
P
(
6
)
=

e

!

(b) The probability, rounded to 4 decimal places, is

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here are the solutions to your probability problems:
**1. Binomial Probability (n=18, k=4, p=0.41):**
The binomial probability formula is:
P(k) = (nCk) * p^k * (1-p)^(n-k)
Where nCk is the binomial coefficient, calculated as n! / (k! * (n-k)!).
P(4) = (18C4) * (0.41)^4 * (1 - 0.41)^(18-4)
P(4) = 3060 * 0.028257761 * 0.006553568
P(4) ≈ 0.0567
**2. Binomial Probability (n=11, k=7, p=2/3):**
P(7) = (11C7) * (2/3)^7 * (1 - 2/3)^(11-7)
P(7) = 330 * (128/2187) * (1/81)
P(7) ≈ 0.2376
**3. Binomial Probability Distribution (n=2, 7 cameras, 4 defective):**
Since the cameras are selected *with* replacement, we use the binomial distribution. The probability of selecting a defective camera is 4/7, and the probability of selecting a non-defective camera is 3/7.
a. **Binomial Probability Distribution for X:**
* P(X=0) = (2C0) * (4/7)^0 * (3/7)^2 = 1 * 1 * 9/49 ≈ 0.18
* P(X=1) = (2C1) * (4/7)^1 * (3/7)^1 = 2 * 4/7 * 3/7 = 24/49 ≈ 0.49
* P(X=2) = (2C2) * (4/7)^2 * (3/7)^0 = 1 * 16/49 * 1 = 16/49 ≈ 0.33
b. **Expected Value of X:**
E(X) = n * p = 2 * (4/7) = 8/7 ≈ 1.14
**4. Probability of Fewer Than 4 Tornadoes in 23 Years:**
Since tornadoes are independent and the probability of a tornado in a year is constant, we can model this using the binomial distribution with n = 23 and p = 0.11. We want P(X < 4), which is P(X=0) + P(X=1) + P(X=2) + P(X=3).
Using a binomial calculator or software is highly recommended for this. Calculating each term and summing them, we get:
P(X < 4) ≈ 0.0785 + 0.2022 + 0.2606 + 0.2103 ≈ 0.7516
**5. Poisson Probability (λ=7, k=6):**
a. **Poisson Probability Formula Details:**
P(6) = (7^6 * e^-7) / 6!
b. **Probability:**
P(6) = (117649 * 0.00091188) / 720
P(6) ≈ 0.1490

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