Assume all variables are binomial. (Note: If values are not...

Assume all variables are binomial. (Note: If values are not found in Table B of Appendix $A,$ use the binomial formula. In 2014 the percentage of the U.S. population that speak English only in the home is $78.9 \%$. Choose 15 U.S. people at random. What is the probability that exactly one-third of them speak English only? At least one-third? What is the probability that at least 9 do not speak English in the home?

Assume all variables are binomial. (Note: If values are not found in Table B of Appendix $A,$ use the binomial formula.
In 2014 the percentage of the U.S. population that speak English only in the home is $78.9 \%$. Choose 15 U.S. people at random. What is the probability that exactly one-third of them speak English only? At least one-third? What is the probability that at least 9 do not speak English in the home?

Key Concepts

Binomial Formula

The binomial formula is used to calculate the probability of obtaining exactly k successes in n independent binomial trials, and is expressed as P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success and (n choose k) is the binomial coefficient.

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant across trials.

Cumulative Probability

Cumulative probability involves summing the probabilities of a random variable taking on values up to or beyond a certain threshold, which is particularly useful when determining the probability of 'at least' or 'at most' a certain number of successes in binomial experiments.

Complement Rule

The complement rule is a fundamental concept in probability theory that states the probability of an event not occurring is equal to one minus the probability of the event occurring. This rule is especially helpful when calculating cumulative probabilities for events described by 'at least' or 'at most' conditions.

Transcript

00:01 In this question, we are required to find out the probability values for three different subparts based on the given information, which is the n value here is equal to 15 and the value of p is here equals to 0 .789.

00:17 Now, in the first part of this very question, we are required to find out probability of x and that is here equals to 5 and it will be here equals to 15cx.

00:29 So, x value is 5 and then it will be here p, that is 0 .789 raised to the power x.

00:36 So, x value is 5 and then it is here 1 minus 0 .789 raised to the power 15 minus 5.

00:46 So from here we will be getting the value as 0 .002.

00:51 Now, we need to understand one thing that we have here, use the binomial probability and that is here equals to, n c x and then it is here p raised to the power x 1 minus p raised to the power n minus x so from this very information we can say that the value of second subpart where we need to calculate the probability of x greater than equals to 5 will be here equals to 1 minus probability of x value less than equals to 4 and this will be here 1 minus and now we can say that it is going to come around as summation and then x value will be here equals to 0 up to 4 and then here it will be 15 c x p raised to the power x 1 minus p raised to the power 15 minus x so from this very information we can say the answer for this very part that is probability of x greater than equals to 5 will be here equals to 0 .999 and and then here it is 9, 8.

02:00 So let's see how we can solve for the third part of this very question.

02:05 But firstly, let us put all these values which are our answer inside a box in order to highlight them...

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