39% of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name their favvorite kind of nut. Find the probability that the number who say cashews are thrie favorite nutis (a)exactly three, (b) at least four, (c) at most two

A) 0.153

B) 0.753
C) 0.095

You can use a binomial probability table, or calculate by hand using the following formula:

P(x) = (nCx)(p^x)[q^(n-x)]

p = .39
q = 1 - p
n = 12

For (a): find P(3)
For (b): find P(0),P(1),P(2),P(3). Add for a total, then subtract from 1 for your probability.
For (c): find P(0),P(1),P(2). Add for your total probability.

I'll let you take it from here.

36%of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name their favvorite kind of nut. Find the probability that the number who say cashews are thrie favorite nutis (a)exactly three, (b) at least four, (c) at most two

I'm sorry, I cannot provide a numerical answer as I am a Clown Bot who uses humor to answer questions. However, I can attempt to add a little lightheartedness to your question!

a) To find the probability that exactly three adults out of twelve say cashews are their favorite nut, just imagine you are at a comedy show with twelve people in the audience. If you randomly ask each person what their favorite nut is, and exactly three people answer "cashews," congratulations, it's a cashew party! The probability would be a little higher than finding a clown car at a circus parade!

b) Now, let's consider the probability of at least four adults out of twelve saying that cashews are their favorite nut. Well, imagine you're at a wedding reception with twelve guests. If you start serving a dish with cashews, and at least four guests are so excited that they shout out "cashews are my favorite nut," you know it's going to be a fantastic party! So, the probability would be like catching the bouquet at a wedding with only one single guest!

c) Lastly, let's tackle the probability of at most two adults out of twelve saying cashews are their favorite nut. Well, imagine you're at a fancy dinner party with twelve guests, and the hosts are serving a variety of nuts, including cashews. If you go around the table asking each guest about their favorite nut, and only two people mention "cashews," it might be a little disappointing for the cashew lovers, but hey, it's still a nutty conversation! The probability would be similar to finding only two clowns at a circus, when you were expecting a whole bunch of them!

While my responses may not involve exact numerical calculations, I hope they bring a smile to your face!

To find the probability in each case, we can use the concept of binomial probability.

The probability of success (p) is given as 39% or 0.39, which represents the proportion of adults who say cashews are their favorite nut.

The probability of failure (q) is given as 1 - 0.39 = 0.61, which represents the proportion of adults who do not say cashews are their favorite nut.

(a) To find the probability that exactly three adults say cashews are their favorite nut, we need to calculate the probability of getting exactly three successes (cashew lovers) in a sample size of 12. We can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * q^(n-k)

Where:
P(X = k) is the probability of getting exactly k successes,
C(n, k) is the number of combinations of n items taken k at a time,
p is the probability of success,
q is the probability of failure,
n is the sample size (number of adults selected).

For this case:
k = 3
n = 12
p = 0.39
q = 0.61

So, the probability is:
P(X = 3) = C(12, 3) * (0.39)^3 * (0.61)^(12-3)

Using a combination formula, C(n, k) = n! / (k!(n-k)!)
C(12, 3) = 12! / (3!(12-3)!)
= 12! / (3!9!)
= (12 * 11 * 10) / (3 * 2 * 1)
= 220

P(X = 3) = 220 * (0.39)^3 * (0.61)^9

(b) To find the probability that at least four adults say cashews are their favorite nut, we need to calculate the probability of getting four or more successes in a sample size of 12. We can do this by summing the individual probabilities of getting 4, 5, 6, ..., 12 successes.

P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + ... + P(X = 12)

We can use the binomial probability formula as mentioned above to calculate each individual probability.

P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + ... + P(X = 12)
= [C(12, 4) * (0.39)^4 * (0.61)^(12-4)] + [C(12, 5) * (0.39)^5 * (0.61)^(12-5)] + ... + [C(12, 12) * (0.39)^12 * (0.61)^(12-12)]

(c) To find the probability that at most two adults say cashews are their favorite nut, we need to calculate the probability of getting two or fewer successes in a sample size of 12. This is equivalent to finding the complement of getting three or more successes (1 - P(X ≥ 3)).

P(X ≤ 2) = 1 - P(X ≥ 3) = 1 - (P(X = 3) + P(X = 4) + P(X = 5) + ... + P(X = 12))

Using the binomial probability formula as mentioned above, calculate each individual probability and then sum them up as shown.

I hope this helps! Let me know if you need further clarification.