Question

39​% of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name his or her favorite nut. Find the probability that the number who say cashews are their favorite nut is​ (a) exactly​ three, (b) at least​ four, and​ (c) at most two. If​ convenient, use technology to find the probabilities

Answer 1

To find the probabilities, the binomial probability formula is used, where for (a) we calculate for exactly 3 adults, for (b) we calculate the complement of having at most 3 adults, and for (c) we sum the probabilities of having 0, 1, or 2 adults choosing cashews as their favorite nut.

Step-by-step explanation:

To solve this probability question, we will use the binomial probability formula which is given by:

P(X = k) = (n choose k) *
`p^k` *
`(1 - p)^{(n-k)`

where:

• n is the number of trials (in this case, 12 adults)
• k is the number of successes (number of adults choosing cashews as their favorite nut)
• p is the probability of success on an individual trial (39%), or 0.39

Part (a): Exactly Three

P(X = 3) = (12 choose 3) *
`0.39^3` *
`0.61^9`

Part (b): At Least Four

P(X ≥ 4) = 1 - P(X ≤ 3), which is 1 minus the sum of the probabilities of having 0, 1, 2, or 3 adults choosing cashews.

Part (c): At Most Two

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)