Math  /  Data & Statistics

QuestionA local fast-food restaurant is running a "Draw a three, get it free" lunch promotion. After each customer orders, a touch-screen display shows the message, "Press here to win a free lunch." A computer program then simulates one card being drawn at random from a standard deck of playing cards. If the chosen card is a 3, the customer's order is free. (Note that the probability of drawing a 3 from a standard deck of playing cards is 4/524 / 52.) Otherwise, the customer must pay the bill. Suppose that 250 customers place lunch orders on the first day of the promotion. Let XX == the number of people who win a free lunch. Explain why X is a binomial random variable. B- choose your answer... \square "failure"=Anything but a type your answer... \square - "success"=Draw a type your answer... \square \qquad - Knowing whether or not one person gets a type your answer... \square tells you choose your answer... \square about whether or not 11- choose your answer... \square another person gets a type your answer... \square N- \square n=-n= type your answer... \square
S- \square p=p= type your answer... \square

Studdy Solution

STEP 1

What is this asking? We need to explain why the number of people winning free lunch is a binomial random variable, and then identify the number of trials, the probability of success, and whether trials are independent. Watch out! Don't mix up the number of trials with the total number of customers.
Also, remember that a binomial random variable counts the *number of successes*, not the probability of success.

STEP 2

1. Define Binomial Random Variable
2. Identify BINS
3. Calculate Probability and Number of Trials

STEP 3

A **binomial random variable** counts the number of successes in a fixed number of independent trials.
Each trial has only two possible outcomes: success or failure.
The probability of success is the same for each trial.
Does that sound familiar?
Let's see!

STEP 4

Each customer either wins a free lunch (**success**) or doesn't (**failure**).
Two outcomes!
So far, so good!

STEP 5

Whether one customer wins a free lunch doesn't affect whether another customer wins.
The card draws are independent!
This is crucial for our binomial setup.

STEP 6

There are **250** customers, so there are **250** trials. n=250n = 250.
Each customer gets one "trial" to win a free lunch.

STEP 7

The probability of drawing a 3 (winning a free lunch) is always 452\frac{4}{52}, which simplifies to 113\frac{1}{13}.
This probability stays the same for each customer. p=113p = \frac{1}{13}.

STEP 8

The probability of success (pp) is the probability of drawing a 3, which is 452=1130.0769\frac{4}{52} = \frac{1}{13} \approx 0.0769.
So, there's about a **7.69%** chance of any single customer winning a free lunch.

STEP 9

The number of trials (nn) is the number of customers, which is **250**.

STEP 10

XX is a binomial random variable because it counts the number of successes (free lunches) in 250 independent trials, each with a probability of success of 113\frac{1}{13}. - "success"= Draw a 3 - "failure"= Anything but a 3 - Knowing whether or not one person gets a free lunch tells you *nothing* about whether or not another person gets a free lunch. - n=250n = 250 - p=113p = \frac{1}{13}

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