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. Fiftythree percent of all persons in the U.S. population have at least some college education. Choose 10 persons at random. Find the probability that a. Exactly one-half have some college education b. At least 5 do not have any college education c. Fewer than 5 have some college education

Assume all variables are binomial. (Note: If values are not found in Table B of Appendix $A,$ use the binomial formula.
Fiftythree percent of all persons in the U.S. population have at least some college education. Choose 10 persons at random. Find the probability that
a. Exactly one-half have some college education
b. At least 5 do not have any college education
c. Fewer than 5 have some college education

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is fundamental to situations where outcomes can be categorized in a binary way, such as having some college education or not.

Combination Formula

The combination formula, often expressed as C(n, k) = n! / (k!(n-k)!), is used to determine the number of ways to choose k successes out of n independent trials. This concept is crucial in calculating binomial probabilities, as it accounts for the different ways a specified number of successes can occur.

Probability Mass Function (PMF) of the Binomial Distribution

The binomial probability mass function, given by P(X = k) = C(n, k) * p^k * (1-p)^(n-k), is used to calculate the probability of exactly k successes in n independent trials with success probability p. It provides a systematic way to compute exact event probabilities in binomial settings.

Cumulative Probability Calculations

Cumulative probability involves adding the probabilities of multiple outcomes to determine the likelihood of an event occurring within a range, such as 'at least' or 'fewer than' a certain number of successes. This concept is key when the problem requires summing probabilities for several outcomes rather than just one exact outcome.

Transcript

00:01 In this question we are required to solve three sub -problems based on the given information that n is equal to 10 and then p value is here equals to 0 .53 and now we must know the formula for binomial probability and it can be here written as n and then here it is k and then it will be here p raised to the power k 1 minus p raised to the power n minus k.

00:29 So from this point very information we can say that first part is here basically about probability of x is equals to 5 so it will be here 10 and then here it is 5 and now it will be here 0 .53 raised to the power 5 and then it is here 1 minus 0 .53 and then it will be here raised to the power 10 minus 5 so that is here equals to 0 .2417 now let's see how we can calculate the answer for the second part of this very question.

01:05 So in the second part, we are required to find out probability of x greater than equals to 5 and that is here equals to summation and then we can say that it will be i and it is going to be equals to 5 and it will go on up to 10 and then it will be here we can say that it is and then here it is i and now it will be here p raised to the power i and then it will be here that 1 minus p raised to the power n minus i that is n value is 10 and then here it is i so based on this very information it can be here written that the value of probability of x greater than equals to 5 will be here equals to probability of 5 plus probability of 5 plus probability of 6 and it will be here plus probability of 7 and then it is here plus probability of 8 and then it is here plus probability of 9 and at last it will be here probability of 10.

02:09 So this very value is here going to come around.

02:13 Let us write it and it will be here equals to 0.

02:17 We can say 69, 4, 3, 2 as our answer...

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