In a recent survey, 95 % of the students in a certain...

In a recent survey, $95 \%$ of the students in a certain university have cell phones. In a sample of 30 students, what is the probability? a. that all 30 students have cell phones b. that less than 4 students have cell phones c. that more than 22 students have cell phones d. that at least 20 students have cell phones

In a recent survey, $95 \%$ of the students in a certain university have cell phones. In a sample of 30 students, what is the probability?
a. that all 30 students have cell phones
b. that less than 4 students have cell phones
c. that more than 22 students have cell phones
d. that at least 20 students have cell phones

Key Concepts

Cumulative Probabilities and Complement Rule

Cumulative probabilities in the context of the binomial distribution involve summing the probabilities from the PMF over a range of outcomes to find the probability of obtaining at most (or at least) a certain number of successes. The complement rule is often applied to simplify the calculation of tail probabilities, such as finding the probability of more than or less than a given number of successes.

Probability Mass Function (PMF) for Binomial Distribution

The probability mass function for a binomial distribution provides the probability of obtaining exactly k successes in n independent trials. It uses the binomial coefficient to count the number of ways k successes can occur and multiplies it by the success probability raised to the k-th power and the failure probability raised to the (n-k)th power.

Binomial Distribution

The binomial distribution is a probability model that describes the number of successes in a fixed number of independent experiments or trials (known as Bernoulli trials), where each trial has only two possible outcomes (success or failure) and a constant probability of success. It is widely used in calculating the likelihood of a specific number of successes over a given sample size.

Transcript

00:01 So this is a binomial distribution where n is going to be 30 students, 95 % is your probability, and then your x is the number of successes.

00:10 It changes from problem to problem.

00:12 So on part a, they're asking the probability that all 30 students have a cell phone.

00:19 And so there's a binomial pdf button on your calculator that finds exactly 30, and that's what we want here.

00:29 If you go under second or shift, vars, v -a -r -s, on the ti -84 and scroll, so you find the binomial pdf button.

00:37 What you do from there is you put in n, 30.

00:40 The probability is 0 .95.

00:44 And then in this case, we're looking for all 30 students having a cell phone.

00:50 And so that answer comes out to 0 .21, or 21 % that all of them would have it, all 30.

00:57 Fairly likely because 95 % of them say they have it.

01:01 The next one says less than four.

01:03 So i kind of drew out the different possibilities.

01:05 You could have zero students having cell phones all the way up to 30.

01:09 So we're talking about these students, less than four would be 0, 1, 2, or 3.

01:14 So rather than doing a pdf of 0, 1, 2, and 3 and adding them together, there's a binomial cdf.

01:22 The c stands for cumulative button that it will just automatically add all those up if i put a three in.

01:29 So i wanna go again to second vars vrs and then find the binomial cdf button.

01:35 And i'd put in 30 and then 0 .95 and then three.

01:42 If i put in three, it will get me all the ones for 0, 1, 2, and 3 and add them up.

01:48 When you do that though, you virtually get zero.

01:51 You end up with a scientific notation with like 25 zeros.

01:57 And if you think about it, 95 % of the students have cell phones...

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