31% of adults say that cashews are their favorite kind of nut. You randomly select 12 adults and ask them to name their favorite nut. Find the probability that the number who say cashew istheir favorite nut is a(exactly three) b. at least four and c at most 2
a. 12C3(.31)^3(.69)^9 = .23235
b. 1-[(12C0)(.31)^0(.69)^12 + 12C2(.31)^2(.69)^10 + 12C3(.31)^3(.69)^9 ] = 0.5381
c. 12C0(.31)^0 (.69)^12+ 12C1(.31)^1(.69)^11 + 12C2(.31)^2(.69)^10 = 0.2295875
To solve this problem, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (n C k) * p^k * q^(n-k)
where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials
- p is the probability of success on each trial
- q is the probability of failure on each trial (1 - p)
- (n C k) is the binomial coefficient, which is the number of ways to choose k successes from n trials
Now let's calculate the probabilities:
a) Exactly three people say that cashews are their favorite nut.
n = 12 (total number of adults)
k = 3 (number of successes)
p = 0.31 (probability that an adult says cashew is their favorite nut)
q = 1 - p = 1 - 0.31 = 0.69 (probability of failure)
P(X = 3) = (12 C 3) * (0.31)^3 * (0.69)^(12-3)
P(X = 3) = (220) * (0.31)^3 * (0.69)^9
b) At least four people say that cashews are their favorite nut.
We need to calculate the probability of four, five, six, ..., twelve people saying cashews are their favorite nut and then sum them up.
P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + ... + P(X = 12)
To calculate these individual probabilities, we can follow the same formula as in part a.
c) At most two people say that cashews are their favorite nut.
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
Again, we can calculate these individual probabilities using the same formula.
Please note that you need to substitute the values of n, k, p, and q into the binomial probability formula to calculate each probability.
To find the probability in each case, we need to first determine the number of ways each situation can occur and divide it by the total number of possible outcomes.
a) Probability of exactly three adults saying cashews are their favorite nut:
We have randomly selected 12 adults, and the probability that any particular adult says cashews are their favorite nut is 31%. To find the probability of exactly three adults, we will use the binomial probability formula:
P(X = k) = (n C k) * (p^k) * (q^(n-k))
where:
P(X = k) is the probability of getting exactly k successes,
n is the number of trials (sample size),
k is the number of successes,
p is the probability of success in one trial,
q is the probability of failure in one trial.
In this case:
n = 12 (sample size)
k = 3 (number of adults saying cashews)
p = 0.31 (probability of saying cashews)
q = 1 - p = 1 - 0.31 = 0.69 (probability of not saying cashews)
Now we can calculate the probability:
P(X = 3) = (12 C 3) * (0.31^3) * (0.69^(12-3))
(12 C 3) = 12! / (3!(12-3)!)
= 12! / (3!9!)
= (12 * 11 * 10) / (3 * 2 * 1)
= 220
P(X = 3) = 220 * (0.31^3) * (0.69^9)
≈ 0.2554
Therefore, the probability of exactly three adults saying cashews are their favorite nut is approximately 0.2554.
b) Probability of at least four adults saying cashews are their favorite nut:
We can approach this by finding the probabilities of four, five, six, ..., twelve adults saying cashews and adding them up.
P(at least four) = P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + ... + P(X = 12)
We already calculated P(X = 3) in part a, so we can subtract it from 1 to find the sum of probabilities:
P(at least four) = 1 - P(X = 0) - P(X = 1) - P(X = 2) - P(X = 3)
To calculate these probabilities, we repeat the binomial probability formula with different values of k for each probability.
P(X = 0) = (12 C 0) * (0.31^0) * (0.69^12)
P(X = 1) = (12 C 1) * (0.31^1) * (0.69^11)
P(X = 2) = (12 C 2) * (0.31^2) * (0.69^10)
Using the same calculation method as in part a, we find:
P(X = 0) ≈ 0.1125
P(X = 1) ≈ 0.2856
P(X = 2) ≈ 0.3288
Substituting these values into the formula:
P(at least four) = 1 - 0.1125 - 0.2856 - 0.3288 - 0.2554
≈ 0.0177
Therefore, the probability of at least four adults saying cashews are their favorite nut is approximately 0.0177.
c) Probability of at most two adults saying cashews are their favorite nut:
To find this probability, we need to calculate the probabilities of exactly zero, exactly one, and exactly two adults saying cashews, and sum them up:
P(at most two) = P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
We already calculated P(X = 0), P(X = 1), and P(X = 2) in part b. Substituting those values:
P(at most two) = 0.1125 + 0.2856 + 0.3288
≈ 0.727
Therefore, the probability of at most two adults saying cashews are their favorite nut is approximately 0.727.