Mathematics, 16.07.2019 14:30 mishcacross21

36% 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

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Answer from: mooreadrian41235

a) 18.49% probability that the number who say cashews are their favorite nut is exactly three.

b) 67.99% probability that the number who say cashews are their favorite nut is at least four.

c) 13.52% probability that the number who say cashews are their favorite nut is at most two.

Step-by-step explanation:

For each adult, there are only two possible outcomes. Either cashews are their favorite kind of nut, or they are not. The probability of an adult having cashews as their favorite kind of nut is independent of other adults. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

36% of adults say cashews are their favorite kind of nut.

This means that p = 0.36

12 adults

This means that n = 12

(a) exactly three

This is P(X = 3).

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 3) = C_{12,3}.(0.36)^{3}.(0.64)^{9} = 0.1849$

18.49% probability that the number who say cashews are their favorite nut is exactly three.

(b) at least four

Either there are less than four, or there are at least four. The sum of the probabilities of these events is decimal 1. So

$P(X < 4) + P(X \geq 4) = 1$

We want $P(X \geq 4)$ . So

$P(X \geq 4) = 1 - P(X < 4)$

In which

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{12,0}.(0.36)^{0}.(0.64)^{12} = 0.0047$

$P(X = 1) = C_{12,1}.(0.36)^{1}.(0.64)^{11} = 0.0319$

$P(X = 2) = C_{12,2}.(0.36)^{2}.(0.64)^{10} = 0.0986$

$P(X = 3) = C_{12,3}.(0.36)^{3}.(0.64)^{9} = 0.1849$

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0047 + 0.0319 + 0.0986 + 0.1849 = 0.3201

$P(X \geq 4) = 1 - P(X < 4)[ = 1 - 0.3201 = 0.6799$

67.99% probability that the number who say cashews are their favorite nut is at least four.

c) At most two

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{12,0}.(0.36)^{0}.(0.64)^{12} = 0.0047$

$P(X = 1) = C_{12,1}.(0.36)^{1}.(0.64)^{11} = 0.0319$

$P(X = 2) = C_{12,2}.(0.36)^{2}.(0.64)^{10} = 0.0986$

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0047 + 0.0319 + 0.0986 = 0.1352$

13.52% probability that the number who say cashews are their favorite nut is at most two.

Answer from: Quest

image result for what is another name for consecutive interior angles?

when two lines are crossed by another line (called the transversal): the pairs of angles on one side of the transversal but inside the two lines are called consecutive interior angles. in this example, these are consecutive interior angles: d and f.

step-by-step explanation:

Answer from: Quest

step-by-step explanation:

you can add the two numbers and divide the result by 2. or you can find the gap between the two numbers and add half of this to the smaller number.