Mathematics: The random variable X, the number of people to revert to their first learned method under stress, has a binomial distribution with n = 40 and p = 0.2. Find the probability that at least eight of the 40 subjects revert to their first learned method. Round your answer to four decimal places.

Step-by-step explanation:the correct answer is B on Edgen. :)...

The random variable X, the number of people to

Ответ:

lovebunny33921

19.01.2022

Mathematics

0.5628 = 56.28% probability that at least eight of the 40 subjects revert to their first learned method.

Step-by-step explanation:

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.

In this problem we have that:

n = 40, p = 0.2

Find the probability that at least eight of the 40 subjects revert to their first learned method.

Either less than 8 do, or at least 8 do. The sum of the probabilities of these events is decimal 1. So

$P(X < 8) + P(X \geq 8) = 1$

We want $P(X \geq 8)$ . So

$P(X \geq 8) = 1 - P(X < 8)$

In which

P(X < 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)

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

$P(X = 0) = C_{40,0}.(0.2)^{0}.(0.8)^{40} = 0.0001$

$P(X = 1) = C_{40,1}.(0.2)^{1}.(0.8)^{39} = 0.0013$

$P(X = 2) = C_{40,2}.(0.2)^{2}.(0.8)^{38} = 0.0065$

$P(X = 3) = C_{40,3}.(0.2)^{3}.(0.8)^{37} = 0.0205$

$P(X = 4) = C_{40,4}.(0.2)^{4}.(0.8)^{36} = 0.0475$

$P(X = 5) = C_{40,5}.(0.2)^{5}.(0.8)^{35} = 0.0854$

$P(X = 6) = C_{40,6}.(0.2)^{6}.(0.8)^{34} = 0.1246$

$P(X = 7) = C_{40,7}.(0.2)^{7}.(0.8)^{33} = 0.1513$

P(X < 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) = 0.0001 + 0.0013 + 0.0065 + 0.0205 + 0.0475 + 0.0854 + 0.1246 + 0.1513 = 0.4372

$P(X \geq 8) = 1 - P(X < 8) = 1 - 0.4372 = 0.5628$

0.5628 = 56.28% probability that at least eight of the 40 subjects revert to their first learned method.

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The random variable X, the number of people to revert to their first learned method under stress, has a binomial distribution with n = 40 and p = 0.2. Find the probability that at least eight of the 40 subjects revert to their first learned method. Round your answer to four decimal places.

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