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Chapter 16: Problem 5

LeBron again Let's take one last look at the LeBron James picture search. You know his picture is in \(20 \%\) of the cereal boxes. You buy five boxes to see how many pictures of LeBron you might get. a) Describe how you would simulate the number of pictures of LeBron you might find in five boxes of cereal. b) Run at least 30 trials. c) Based on your simulation, estimate the probabilities that you get no pictures of LeBron, 1 picture, 2 pictures, etc. d) Find the actual probability model. e) Compare the distribution of outcomes in your simulation to the probability model.

Short Answer

Expert verified
Simulate finding pictures using random numbers for 30 trials. Use a binomial model for the actual probabilities and compare them to see the resemblance or difference.

Step by step solution

01

Understanding the Scenario

We're determining how many LeBron pictures we might find in 5 cereal boxes, given that there's a 20% chance of finding one in each box. We'll be simulating this process and comparing our simulated results to theoretical probabilities.
02

Simulating the Scenario

To simulate the scenario, we can use a random number generator where a number from 1 to 5 means no picture ('failure'), and 6 means a picture of LeBron ('success'). Generate five random numbers for each trial to represent the five cereal boxes.
03

Conducting 30 Trials

Perform this simulation 30 times, where each trial consists of examining five boxes. For each trial, count how many 'successes' (LeBron pictures) you have. Record the result of each trial.
04

Estimating Probabilities based on Simulation

Count how many trials resulted in 0, 1, 2, 3, 4, and 5 pictures of LeBron. Divide each count by 30 to estimate the probabilities of finding each number of pictures from your simulation results.
05

Constructing the Probability Model

The probability model is a binomial distribution with parameters \(n = 5\) and \(p = 0.2\). The probability of finding exactly \(k\) pictures of LeBron in 5 boxes is given by the binomial formula: \(P(X=k) = \binom{5}{k} \times (0.2)^k \times (0.8)^{5-k}\) for \(k = 0, 1, 2, 3, 4, 5\). Calculate these probabilities.
06

Comparing Results

Compare the probabilities obtained from the simulation to the theoretical probabilities from the binomial model. Observe any differences or similarities in the distribution of outcomes to understand the accuracy and reliability of the simulation compared to theoretical probabilities.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution
When we talk about a binomial distribution, we're discussing a specific probability model. It's perfect for answering questions like, "What's the chance of getting a certain number of successes in a set number of trials?" In our exercise, this translates to finding LeBron pictures in cereal boxes.

The binomial distribution is characterized by two parameters:
  • \(n\), the number of trials, which in this case is 5, representing the 5 cereal boxes.
  • \(p\), the probability of success on each trial. Here, that success is finding a LeBron picture, with a probability of 0.2 or 20\%.
We can use a specific formula to calculate the probability of exactly \(k\) successes, or pictures, which involves the combination formula \(\binom{n}{k}\), indicating the number of ways to choose \(k\) successes in \(n\) trials. This gives us a way to theoretically model and understand the problem.
Random Number Generation
Random number generation is a crucial part of running simulations, such as one for estimating how many LeBron pictures are in the cereal boxes. Instead of physically counting pictures in boxes, a computer or a calculator can generate random numbers to simulate the process.

Here's how it works in a simple way:
  • Assign numbers 1-5 as 'failures' (no LeBron picture).
  • Assign the number 6 as a 'success' (one LeBron picture).
By creating this setup, we simulate the opening of each cereal box by generating a random number. This replication of the real-world process allows for quick and repetitive experiments which are useful in probability exercises.
Probability Estimation
Once we've conducted our simulation, we can use the results to estimate probabilities. This is a straightforward process involving our findings from multiple trials.

Here鈥檚 a step-by-step on how to do it:
  • Count how many trials resulted in 0 LeBron pictures, 1 picture, 2 pictures, etc.
  • Divide that count by the total number of trials (30, in this exercise).
For example, if 6 out of 30 trials resulted in finding 2 LeBron pictures, the estimated probability of finding exactly 2 pictures is \( \frac{6}{30} = 0.2 \). This method gives us an empirical approximation of the probability distribution observed in the simulation trials.
Probability Models
Probability models, like the binomial distribution, provide a theoretical explanation of our random process. These models are built using mathematical formulas, predicting the probability of various outcomes under defined conditions.

In our exercise, the probability model uses the binomial distribution for \(n = 5\) and \(p = 0.2\). Using the formula \(P(X=k) = \binom{5}{k} \times (0.2)^k \times (0.8)^{5-k}\), we calculate the exact probability for each potential number of LeBron pictures鈥攔anging from 0 to 5.
  • This model helps in understanding the "expected" distribution of our simulation and determines how close our simulated results align with theoretical expectations.
By comparing the simulation to this model, we can assess how well our random number generation and experimental probabilities reflect real-world scenarios.

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Most popular questions from this chapter

No-shows An airline, believing that \(5 \%\) of passengers fail to show up for flights, overbooks (sells more tickets than there are seats). Suppose a plane will hold 265 passengers, and the airline sells 275 tickets. What's the probability the airline will not have enough seats, so someone gets bumped?

Hoops A basketball player has made \(80 \%\) of his foul shots during the season. Assuming the shots are independent, find the probability that in tonight's game he a) misses for the first time on his fifth attempt. b) makes his first basket on his fourth shot. c) makes his first basket on one of his first 3 shots.

\- Apples An orchard owner knows that he'll have to use about \(6 \%\) of the apples he harvests for cider because they will have bruises or blemishes. He expects a tree to produce about 300 apples. a) Describe an appropriate model for the number of cider apples that may come from that tree. Justify your model. b) Find the probability there will be no more than a dozen cider apples. c) Is it likely there will be more than 50 cider apples? Explain.

\- Hot hand A basketball player who ordinarily makes about \(55 \%\) of his free throw shots has made 4 in a row. Is this evidence that he has a "hot hand" tonight? That is, is this streak so unusual that it means the probability he makes a shot must have changed? Explain.

Chips Suppose a computer chip manufacturer rejects \(2 \%\) of the chips produced because they fail presale testing. a) What's the probability that the fifth chip you test is the first bad one you find? b) What's the probability you find a bad one within the first 10 you examine?
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