In a recent study, 35 % of people surveyed indicated...

In a recent study, $35 \%$ of people surveyed indicated chocolate was their favorite flavor of ice cream. Suppose we select a sample of 10 people and ask them to name their favorite flavor of ice cream. a. How many of those in the sample would you expect to name chocolate? b. What is the probability exactly four of those in the sample name chocolate? c. What is the probability four or more name chocolate? $?$ ?

In a recent study, $35 \%$ of people surveyed indicated chocolate was their favorite flavor of ice cream. Suppose we select a sample of 10 people and ask them to name their favorite flavor of ice cream.
a. How many of those in the sample would you expect to name chocolate?
b. What is the probability exactly four of those in the sample name chocolate?
c. What is the probability four or more name chocolate? $?$ ?

Key Concepts

Binomial Probability Formula

The binomial probability formula calculates the likelihood of observing exactly k successes in n independent trials. It is expressed as P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the binomial coefficient representing the number of ways to choose k successes, p is the success probability, and (1-p) is the failure probability.

Cumulative Probability

Cumulative probability in the context of the binomial distribution is the probability of achieving a range of outcomes, such as at least a certain number of successes. This involves summing the individual probabilities for all outcomes within that range, allowing you to compute the likelihood of observing an event at or above a given threshold.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure) and the probability of success remains constant. This distribution is especially useful in scenarios where you're counting the number of times an event occurs, such as determining how many people in a sample choose a particular option.

Expected Value

The expected value in a binomial distribution is the mean number of successes you would anticipate based on the number of trials and the probability of success on each trial. It is calculated with the formula E(X) = n * p, where n is the number of trials and p is the probability of success, providing a measure of central tendency for the distribution.

Transcript

00:01 All right, so a study found that 35 % of people would choose chocolate as their favorite ice cream flavor.

00:09 So if we were to select 10, how many of those in the sample would we expect to name chocolate? well, out of those 10, well, we'd say 10 times 0 .35.

00:25 So three and a half people, so somewhere between 3 and 4 people that we'd expect.

00:29 And the probability is exactly four.

00:35 So i'm going to, before i answer b, i'm going to look at c, because that asks for the probability of four or more.

00:43 So we're going to need, not just four, we need a bunch.

00:46 And since i have a spreadsheet conveniently pulled up here, i'm going to go ahead and make a table of the x values of zero through 10.

01:03 And their probabilities using the binomial probability distribution because each, this is a binomial situation because you could either have chocolate is your favorite or not.

01:23 Two choices, clearly defined.

01:25 We want a clearly defined outcome, four or two or three or something like that, very clearly defined.

01:32 And if someone selects it, it does not have an impact on the other.

01:38 It's independent.

01:40 So it's binomial.

01:42 So that means we can use this formula.

01:45 It's ncx times the probability to the x power times one minus ability.

02:02 That always to the, it's going to be n minus x.

02:09 In this case, n is 10.

02:11 And the probability for each trial is 0 .25...