This week we practice with Binomial Distribution.

You can use Appendix Table E or Excel Function Binomdist.

About 30% of adults in United States have college degree.
(probability that person has college degree is p = 0.30).

If N adults are randomly selected, find probabilities that

1) exactly X out of selected N adults have college degree
2) less than X out of selected N adults have college degree
3) greater than X out of selected N adults have college degree

Choose your numbers for N and X (portion of N).

About 30% of adults in United States have college degree.

(Probability that person has college degree is p = 0.30).

If n adults are randomly selected, find probabilities that:

1) Exactly 24 out of selected 50 adults have college degree
=BINOMDIST(24,50,.30,FALSE) = .0032

2) Less than 36 out of selected 75 adults have college degree
=BINOMDIST(35,75,.03,TRUE) = .9992

3) Greater than 22 out of selected 62 adults have college degree
=1-BINOMDIST(23,62,.03,TRUE) = .0893

Let's choose N = 10 (number of adults randomly selected) and X = 3 (number of adults with college degree).

1) To find the probability that exactly 3 out of 10 selected adults have a college degree, we can use the Binomial Distribution formula:

P(X = k) = C(N, k) * p^k * (1-p)^(N-k)

where:
P(X = k) is the probability of exactly k successes (adults with a college degree),
C(N, k) is the number of combinations of N items taken k at a time (also known as the binomial coefficient),
p is the probability of success (30% or 0.30 in this case), and
N is the number of trials (10 in this case).

Using this formula, we have:

P(X = 3) = C(10, 3) * 0.30^3 * (1-0.30)^(10-3)

Calculating the values, we get:

P(X = 3) = 0.1176

Therefore, the probability that exactly 3 out of 10 selected adults have a college degree is approximately 0.1176.

2) To find the probability that less than 3 out of 10 selected adults have a college degree, we can sum up the probabilities of having 0, 1, and 2 adults with a college degree. Using the same formula as before, the calculations are as follows:

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

P(X = 0) = C(10, 0) * 0.30^0 * (1-0.30)^(10-0)
P(X = 1) = C(10, 1) * 0.30^1 * (1-0.30)^(10-1)
P(X = 2) = C(10, 2) * 0.30^2 * (1-0.30)^(10-2)

Calculating the values, we get:

P(X < 3) = 0.0282 + 0.1211 + 0.2335

P(X < 3) = 0.3828

Therefore, the probability that less than 3 out of 10 selected adults have a college degree is approximately 0.3828.

3) To find the probability that more than 3 out of 10 selected adults have a college degree, we can subtract the probability of having 0, 1, 2, and 3 adults with a college degree from 1 (the total probability). Using the same formula as before, the calculations are as follows:

P(X > 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3))

Calculating the values, we get:

P(X > 3) = 1 - (0.0282 + 0.1211 + 0.2335 + 0.1176)

P(X > 3) = 1 - 0.5004

P(X > 3) = 0.4996

Therefore, the probability that more than 3 out of 10 selected adults have a college degree is approximately 0.4996.

To find the probabilities using the Binomial Distribution, you can use either Appendix Table E or the Excel function Binomdist. I will explain both methods for you.

1) To find the probability that exactly X out of the selected N adults have a college degree:
- Using Appendix Table E:
- Locate the row corresponding to the probability of success, which is p = 0.30.
- Find the column corresponding to the number of trials, which is N.
- The value at the intersection of the row and column represents the probability of getting X successes out of N trials.
- Choose suitable numbers for N (number of adults selected) and X (number of adults with a college degree) to obtain the desired probability.

- Using Excel function Binomdist:
- The syntax for the Binomdist function is BINOMDIST(x, n, p, [cumulative]).
- x represents the number of successes (X in this case).
- n represents the number of trials (N in this case).
- p represents the probability of success (p = 0.30).
- The cumulative argument is optional. If set to TRUE or omitted, it will give you the cumulative probability up to X; if set to FALSE, it will give you the probability of exactly X successes.
- Use suitable values for X and N to calculate the desired probability.

2) To find the probability that less than X out of the selected N adults have a college degree:
- Using Appendix Table E:
- Sum the probabilities for X = 0, 1, 2, ..., X-1.
- This will give you the probability that less than X adults have a college degree.

- Using Excel function Binomdist:
- Set the cumulative argument to TRUE.
- Calculate the cumulative probability for X-1 and subtract it from 1.
- This will give you the probability that less than X adults have a college degree.

3) To find the probability that greater than X out of the selected N adults have a college degree:
- Using Appendix Table E:
- Sum the probabilities for X+1, X+2, ..., N.
- This will give you the probability that more than X adults have a college degree.

- Using Excel function Binomdist:
- Set the cumulative argument to TRUE.
- Calculate the cumulative probability for X and subtract it from 1.
- This will give you the probability that more than X adults have a college degree.

Remember to choose suitable values for N (number of adults selected) and X (number of adults with a college degree) to obtain the desired probabilities.