/*! 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 29 Express all probabilities as fra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 29

Express all probabilities as fractions. As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers between 1 and 75 and, in a separate drawing, you must also select the correct single number between 1 and \(15 .\) Find the probability of winning the jackpot. How does the result compare to the probability of being struck by lightning in a year, which the National Weather Service estimates to be \(1 / 960,000 ?\)

Short Answer

Expert verified
The probability of winning the Mega Millions jackpot is \( \frac{1}{258,890,850} \). It is much less likely than being struck by lightning, whose probability is \( \frac{1}{960,000} \).

Step by step solution

01

Find the Total Number of Possible Combinations of Five Numbers

First, calculate the total number of combinations of selecting 5 numbers out of 75. This is given by the combination formula, which is \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] Here, \( n = 75 \) and \( r = 5 \). So the number of combinations is \[ \binom{75}{5} = \frac{75!}{5!(75-5)!} = \frac{75 \times 74 \times 73 \times 72 \times 71}{5 \times 4 \times 3 \times 2 \times 1} = 17,259,390 \]
02

Calculate the Number of Possible Combinations for the Single Number

Since you need to select 1 correct number between 1 and 15, there are a total of 15 possible choices.
03

Calculate the Total Number of Possible Combinations for Winning the Jackpot

To win the jackpot, you must get both the set of 5 numbers and the single number correct. Therefore, we multiply the number of ways to choose 5 numbers out of 75 by the number of ways to choose 1 number out of 15: \[ 17,259,390 \times 15 = 258,890,850 \]
04

Calculate the Probability of Winning

The probability of winning the jackpot is the ratio of the number of successful outcomes (which is 1, since there is only one winning combination) to the total number of possible outcomes. Thus, the probability is: \[ \frac{1}{258,890,850} \]
05

Compare the Probability of Winning the Jackpot to Being Struck by Lightning

The probability of being struck by lightning in a year is \( \frac{1}{960,000} \). Comparing the two probabilities: \[ \frac{1}{258,890,850} < \frac{1}{960,000} \] This shows that the probability of winning the Mega Millions jackpot is significantly lower than the probability of being struck by lightning.

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.

Combinatorics
Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. In our exercise, we use combinatorics to calculate the number of ways to select 5 numbers out of 75. This is done using the combination formula, denoted as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n \) represents the total number of items, and \( r \) represents the number of items to choose. For instance, to find the number of ways to choose 5 out of 75 numbers, we calculate \( \binom{75}{5} = \frac{75!}{5!(75-5)!} = 17,259,390 \). This tells us that there are 17,259,390 possible combinations of selecting 5 numbers from 75.
Comparative Probability
Comparative probability helps us understand the likelihood of two events occurring relative to each other. In the exercise, we compare the probability of winning the Mega Millions jackpot with being struck by lightning. The lightning strike probability is estimated by the National Weather Service to be \( \frac{1}{960,000} \). By comparing this to the lottery winning probability \( \frac{1}{258,890,850} \), it's clear that winning the lottery is significantly less likely than being struck by lightning. This comparison makes it easy to grasp just how rare winning the jackpot really is.
Statistics Example
The example provided in the exercise is a practical application of statistics. It demonstrates how to calculate and interpret probabilities. We calculate the total number of possible outcomes (258,890,850) and then find the probability of the singular winning outcome, given as \( \frac{1}{258,890,850} \). This real-world example helps illustrate the concepts of probability theory and combinatorics, aiding in better understanding and application.
Probability Theory
Probability theory is the mathematical framework for quantifying uncertainty. One of its primary tools is the calculation of probabilities, which measure the chance that a specific event will occur. In our exercise, we use probability theory to determine the likelihood of winning the lottery, expressed as \( \frac{1}{258,890,850} \). This probability is the ratio of successful outcomes (one winning combination) to the total possible outcomes (258,890,850 combinations). Understanding probability theory helps us make informed predictions about various events, ranging from lottery winnings to weather phenomena.

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

Express all probabilities as fractions. In soccer, a tie at the end of regulation time leads to a shootout by three members from each team. How many ways can 3 players be selected from 11 players available? For 3 selected players, how many ways can they be designated as first, second, and third?

Express all probabilities as fractions. One of the longest words in standard statistics terminology is "homoscedasticity." How many ways can the letters in that word be arranged?

Answer the given questions that involve odds. In the Kentucky Pick 4 lottery, you can place a "straight" bet of \(\$ 1 dollars by selecting the exact order of four digits between 0 and 9 inclusive (with repetition allowed), so the probability of winning is \)1 / 10,000 .$ If the same four numbers are drawn in the same order, you collect \$5000, so your net profit is \$4999. a. Find the actual odds against winning. b. Find the payoff odds. c. The website www.kylottery.com indicates odds of 1: 10,000 for this bet. Is that description accurate?

Find the probability and answer the questions. MicroSort's YSORT gender selection technique is designed to increase the likelihood that a baby will be a boy. At one point before clinical trials of the YSORT gender selection technique were discontinued, 291 births consisted of 239 baby boys and 52 baby girls (based on data from the Genetics \& IVF Institute). Based on these results, what is the probability of a boy born to a couple using MicroSort's YSORT method? Does it appear that the technique is effective in increasing the likelihood that a baby will be a boy?

Redundancy in Hospital Generators Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail \(22 \%\) of the times when they are needed (based on data from Arshad Mansoor, senior vice president with the Electric Power Research Institute). A hospital has two backup generators so that power is available if one of them fails during a power outage. a. Find the probability that both generators fail during a power outage. b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.