Do you take the free samples offered in supermarkets? About...

Do you take the free samples offered in supermarkets? About $60 \%$ of all customers will take free samples. Furthermore, of those who take the free samples, about $37 \%$ will buy what they have sampled. (See reference in Problem $8 .$ ) Suppose you set up a counter in a supermarket offering free samples of a new product. The day you are offering free samples, 317 customers pass by your counter. (a) What is the probability that more than 180 take your free sample? (b) What is the probability that fewer than 200 take your free sample? (c) What is the probability that a customer takes a free sample and buys the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability $P(\text { buy } | \text { sample })=0.37,$ while $P(\text { sample })=0.60$. (d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

Do you take the free samples offered in supermarkets? About $60 \%$ of all customers will take free samples. Furthermore, of those who take the free samples, about $37 \%$ will buy what they have sampled. (See reference in Problem $8 .$ ) Suppose you set up a counter in a supermarket offering free samples of a new product. The day you are offering free samples, 317 customers pass by your counter.
(a) What is the probability that more than 180 take your free sample?
(b) What is the probability that fewer than 200 take your free sample?
(c) What is the probability that a customer takes a free sample and buys the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability $P(\text { buy } | \text { sample })=0.37,$ while $P(\text { sample })=0.60$.
(d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

Key Concepts

Binomial Distribution

This concept models the probability of a given number of successes in a fixed number of independent trials, where each trial has the same probability of success. In problems like this, where each customer independently decides whether to take a sample or purchase a product, the binomial distribution is used to compute the probabilities of various outcomes.

Normal Approximation

For large sample sizes, such as when dealing with hundreds of customers, the binomial distribution can often be approximated by a normal distribution. This approximation simplifies calculations of probabilities for ranges of outcomes, such as finding the probability that the number of customers taking a sample falls within a specific interval.

Multiplication Rule for Dependent Events

This rule is used to determine the probability of two events occurring in sequence, recognizing that the occurrence of the first event affects the probability of the second. It is applied when calculating the probability that a customer both takes a free sample and then buys the product, taking into account the conditional probability of the purchase given the sample.

Conditional Probability

Conditional probability is the likelihood of one event occurring given that another event has occurred. In the context of this problem, it is key to understanding how the probability of a purchase is related to the event of sampling, and it forms the foundation for applying the multiplication rule for dependent events.

Transcript

00:02 Hello! so today we are going to talk about this problem.

00:05 First of all, let's just think what is the inputs of this problem.

00:10 We have a topic counter and we are offering free samples in a grocery store.

00:15 And we have this knowledge that about 60 % of customers will take the free sample.

00:23 So this 60 % is a notation for a mean for us.

00:33 So that's sort of mean of customers getting the sample.

00:51 Okay, so this is the first information.

00:53 Second information is that 37 of those will buy the product.

01:04 We'll buy product.

01:11 And yeah, so this is basically the idea we have.

01:15 And now, at first, we have to know, we know that 317 customers passed by our counter.

01:25 And the question asks these couple of things.

01:29 First, part a.

01:32 What is the probability probability that more than 80 180 people sorry more than 180 people take the sample okay so so, when i see something like this, when i see these couple of keywords, first of all, more than something, and second of all, i have the mean to deal with, i just immediately try to see if i can use markov inequality.

02:26 Inequality.

02:27 So let us remember what was the markov inequality.

02:30 It says that the probability that a random variable x is more than a, for some a, is bounded by the expected value of x divided by that a.

02:54 So let us see what would be the easy match situation.

02:59 We know that so a is given just 180 and we want to know what is the mean of x it's going to be 60 % of x so it's 1 c 700 and 317 times 60 % so the this same probability is going to be equal to this value here.

03:40 I just simplified these two together.

03:44 And yeah, it further simplified us.

03:49 Now, let's move on to part b.

03:53 Part b asks us, what is the probability that fewer than 200 people.

04:14 Take it.

04:18 Again, i look for my keywords.

04:22 Here, i don't have more, i have fewer.

04:25 But remember, it's a probability.

04:27 So the probability of fewer than 200 people take the sample is equal to 1 minus probability more than 200...