/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Binomial Probabilities: Coin Fli... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 15

Binomial Probabilities: Coin Flip A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in Table 3 of Appendix II and compare the table result with the computed result. (a) Find the probability of getting exactly three heads. (b) Find the probability of getting exactly two heads. (c) Find the probability of getting two or more heads. (d) Find the probability of getting exactly three tails.

Short Answer

Expert verified
The probabilities are 0.125 for three heads, 0.375 for two heads, 0.5 for two or more heads, and 0.125 for three tails.

Step by step solution

01

Understand the Binomial Distribution

The problem involves a binomial distribution, as each coin flip is an independent Bernoulli trial with two possible outcomes: heads or tails. The number of trials, \( n \), is 3, and the probability of success (flipping a heads), \( p \), is 0.5.
02

Use the Binomial Formula for (a)

To find the probability of getting exactly three heads, use the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]For \( k = 3 \), we have: \[ P(X = 3) = \binom{3}{3} (0.5)^3 (1-0.5)^{3-3} = 1 \times 0.125 \times 1 = 0.125 \]
03

Use the Binomial Formula for (b)

To find the probability of getting exactly two heads:\[ P(X = 2) = \binom{3}{2} (0.5)^2 (1-0.5)^{3-2} = 3 \times 0.25 \times 0.5 = 0.375 \]
04

Use the Binomial Formula for (c)

For two or more heads (exactly two heads or exactly three heads), add the probabilities:\[ P(X \geq 2) = P(X = 2) + P(X = 3) = 0.375 + 0.125 = 0.5 \]
05

Use the Binomial Formula for (d)

To find the probability of getting exactly three tails, substitute tails as the success (since wanting tails):\[ P(X = 3) = \binom{3}{3} (0.5)^3 (1-0.5)^{3-3} = 1 \times 0.125 \times 1 = 0.125 \]
06

Verify with the Probability Table

Compare each calculated probability (0.125, 0.375, 0.5, 0.125) with the values from Table 3 in Appendix II. Ensure they match.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Binomial Probabilities
When dealing with binomial probabilities, we are essentially calculating the likelihood of a certain number of successes in a set of independent trials. The binomial distribution helps us here. For example, imagine flipping a coin three times. We want to know the probability of getting a specific number of heads.
  • The formula for the binomial probability is:
    \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
  • Here, \( n \) is the total number of trials (three flips), \( k \) is the number of successful outcomes we want (heads in our case), and \( p \) is the probability of success on a single trial.
By calculating each of these, we can determine the probability for each city in our scenario.
Bernoulli Trial
A Bernoulli trial is a fun concept in probability. It's really simple鈥攅ach trial or experiment can only have two possible outcomes. Let's think about flipping a coin as an example. Every time we flip a coin, there are two possible results: heads or tails. That's a Bernoulli trial.
  • The essence of a Bernoulli trial is its binary nature, which means it only has outcomes like success or failure.
  • In our coin flip experiment, getting heads might be considered a success.
These trials are foundational for understanding experiments where each trial is conducted independently, ensuring the results don't affect each other.
Probability Tables
Probability tables are handy tools used to quickly find the likelihood of various outcomes without lengthy calculations. They give us pre-calculated probabilities for specific situations based on variables like the number of trials and probability of success. When dealing with our coin flip example:
  • We can use a probability table to easily verify our manually calculated probabilities.
  • The table lists different probabilities for all possible outcomes based on predefined parameters.
This shortcut is beneficial when checking your work, as it can confirm the answers calculated using the binomial formula.
Coin Flip Experiment
The coin flip experiment is a classic example of binomial probabilities in action. It's simple yet powerful, as it gives a perfect demonstration of basic probability concepts.
  • When we flip a coin three times, like in our problem, each flip is independent.
  • The probability of heads (success) in each flip is 0.5, since it's a fair coin.
The outcomes of such an experiment help us explore a range of probabilities through simple calculations or using helpful resources like probability tables. This hands-on approach solidifies understanding of core probability concepts.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Meteorology: Winter Conditions Much of Trail Ridge Road in Rocky Mountain National Park is over 12,000 feet high. Although it is a beautiful drive in summer months, in winter the road is closed because of severe weather conditions. Winter Wind Studies in Rocky Mountain National Park by Glidden (published by Rocky Mountain Nature Association) states that sustained gale- force winds (over 32 miles per hour and often over 90 miles per hour) occur on the average of once every 60 hours at a Trail Ridge Road weather station. (a) Let \(r=\) frequency with which gale-force winds occur in a given time interval. Explain why the Poisson probability distribution would be a good choice for the random variable \(r\). (b) For an interval of 108 hours, what are the probabilities that \(r=2,3\), and \(4 ?\) What is the probability that \(r<2 ?\) (c) For an interval of 180 hours, what are the probabilities that \(r=3,4\), and \(5 ?\) What is the probability that \(r<3 ?\)

Airlines: Lost Bags USA Today reported that for all airlines, the number of lost bags was May: \(6.02\) per 1000 passengers December: \(12.78\) per 1000 passengers Note: A passenger could lose more than one bag. (a) Let \(r=\) number of bags lost per 1000 passengers in May. Explain why the Poisson distribution would be a good choice for the random variable \(r\). What is \(\lambda\) to the nearest tenth? (b) In the month of May, what is the probability that out of 1000 passengers, no bags are lost? that 3 or more bags are lost? that 6 or more bags are lost? (c) In the month of December, what is the probability that out of 1000 passengers, no bags are lost? that 6 or more bags are lost? that 12 or more bags are lost? (Round \(\lambda\) to the nearest whole number.)

Sociology: Ethics The one-time fling! Have you ever purchased an article of clothing (dress, sports jacket, etc.), worn the item once to a party, and then returned the purchase? This is called a one-time fling. About \(10 \%\) of all adults deliberately do a one-time fling and feel no guilt about it (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press). In a group of seven adult friends, what is the probability that (a) no one has done a one-time fling? (b) at least one person has done a one-time fling? (c) no more than two people have done a one-time fling?

Basic Computation: Binomial Distribution Consider a binomial experiment with \(n=6\) trials where the probability of success on a single trial is \(p=0.85\). (a) Find \(P(r \leq 1)\). (b) Interpretation If you conducted the experiment and got fewer than 2 successes, would you be surprised? Why?

Critical Thinking In an experiment, there are \(n\) independent trials. For each trial, there are three outcomes, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). For each trial, the probability of outcome \(\mathrm{A}\) is \(0.40 ;\) the probability of outcome \(\mathrm{B}\) is \(0.50 ;\) and the probability of outcome \(\mathrm{C}\) is \(0.10 .\) Suppose there are 10 trials. (a) Can we use the binomial experiment model to determine the probability of four outcomes of type A, five of type \(B\), and one of type C? Explain. (b) Can we use the binomial experiment model to determine the probability of four outcomes of type \(\mathrm{A}\) and six outcomes that are not of type \(\mathrm{A}\) ? Explain. What is the probability of success on each trial?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.