44% of adults say cashews are their favorite kind of nut. You randomly select 12 people and ask each to name their favorite nut. Find the probability that the number who say cashews are their favorite nut (a) exactly three, (b) at least four, and (c) at most two.
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To find the probability that exactly three people say cashews are their favorite nut, we can use the binomial probability formula.
The binomial probability formula is given as:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
where:
P(X = k) is the probability of getting exactly k successes
n is the number of trials
k is the number of successes
p is the probability of success in a single trial
In this case, n = 12, k = 3, and p = 0.44 (44% or 0.44 probability of someone saying cashews are their favorite nut).
Using the formula, we can calculate the probability of exactly three people saying cashews:
P(X = 3) = C(12, 3) * (0.44)^3 * (1 - 0.44)^(12 - 3)
To calculate this, we need the combination formula:
C(n, k) = n! / (k! * (n - k)!)
C(12, 3) = 12! / (3! * (12 - 3)!) = 220
Plugging this into the binomial probability formula:
P(X = 3) = 220 * (0.44)^3 * (1 - 0.44)^(12 - 3)
Calculating this expression will give us the probability that exactly three people say cashews are their favorite nut.
To find the probability that at least four people say cashews are their favorite nut, we need to calculate the probabilities for four, five, six, ..., twelve people saying cashews, and sum them up:
P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)
To find the probability that at most two people say cashews are their favorite nut, we need to calculate the probabilities for zero, one, and two people saying cashews, and sum them up:
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
To find the probability in this scenario, we will use the concept of binomial probability.
The probability of success (a person saying cashews as their favorite nut) is given as 44%, which can be expressed as 0.44. The probability of failure (a person not saying cashews as their favorite nut) is the complement of the success probability, which is 1 - 0.44 = 0.56.
The total number of people we randomly selected is 12. We will calculate the probability for each scenario:
(a) Exactly three people say cashews as their favorite nut:
To calculate this probability, we can use the binomial probability formula:
P(X=k) = (n C k) * p^k * (1 - p)^(n - k)
Here, n is the total number of trials (12), k is the number of successful outcomes (3), p is the probability of success (0.44), and (n C k) represents the combination of n and k.
Using the formula:
P(X=3) = (12 C 3) * (0.44)^3 * (0.56)^(12 - 3)
Calculating this expression, you will find the probability.
(b) At least four people say cashews as their favorite nut:
To find this probability, we need to find the probabilities of having 4, 5, 6, 7, 8, 9, 10, 11, and 12 people saying cashews as their favorite nut. Then, we add up these individual probabilities to get the final probability.
P(X >= 4) = P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) + P(X=11) + P(X=12)
To calculate each probability, use the binomial probability formula as described above.
(c) At most two people say cashews as their favorite nut:
To find this probability, we need to find the probabilities of having 0, 1, and 2 people saying cashews as their favorite nut. Then, we add up these individual probabilities.
P(X <= 2) = P(X=0) + P(X=1) + P(X=2)
Again, use the binomial probability formula to calculate each probability.
By following these steps, you should be able to calculate probabilities for each scenario.