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Chapter 11: Problem 9

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to 46 . Use this information to solve Exercises 7-10. Express all probabilities as fractions. A player wins a minimum award of \(\$ 150\) by correctly matching three numbers drawn from white balls ( 1 through 56 ) and matching the number on the gold Mega Ball \({ }^{\mathbb{E}}\) (1 through 46). What is the probability of winning this consolation prize?

Short Answer

Expert verified
The formula \(\frac{\binom{56}{3} \times \binom{46}{1}}{\binom{56}{5} \times \binom{46}{1}}\) gives the probability of winning the consolation prize in this lottery game. Note that the result is a fraction, as requested by the exercise.

Step by step solution

01

Identify the total number of possibilities

Firstly, we need to calculate the total number of possibilities. This can be found by calculating the number of ways to choose 5 numbers out of 56 and 1 number out of 46. Mathematically, this is represented as \(\binom{56}{5} \times \binom{46}{1}\) (combination formula).
02

Calculate the total number of possibilities

The total number of possibilities is calculated by evaluating the expression from step 1. Thus you can use the combination formula, which is defined as \(\frac{n!}{r!(n-r)!}\) where \(n\) is the total number in the set, in this case 56 for the white balls and 46 for the golden balls, and \(r\) is the number we are selecting, in this case 5 for the white balls and 1 for the golden balls.
03

Identify the number of successful outcomes

To win the consolation prize, the player must correctly match three numbers from the white balls and the number on the gold Mega Ball. Thus, the number of successful outcomes can be represented as \(\binom{56}{3} \times \binom{46}{1}\).
04

Calculate the number of successful outcomes

The number of successful outcomes is calculated by evaluating the expression from step 3 using the combination formula, just as we did in step 2.
05

Calculate the probability

Probability is calculated by dividing the number of successful outcomes by the total number of possibilities. Therefore, the probability of winning the consolation prize is \(\frac{\binom{56}{3} \times \binom{46}{1}}{\binom{56}{5} \times \binom{46}{1}}\).

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Key Concepts

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

Combination Formula
Understanding how to use the combination formula is essential in probability problems that involve selecting items from a larger group, such as lottery drawings. In mathematics, the combination formula is used to find the number of ways to choose a subset of items from a larger set where the order does not matter.

For any given set of numbers, the combination formula is represented as \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where:
  • \(n\) represents the total number of items to choose from,
  • \(r\) represents the number of items to select,
  • and \(n!\) (n factorial) is the product of all positive integers up to \(n\).
For example, when playing the lottery and choosing 5 numbers from a set of 56, the combination formula helps determine how many possible selections exist. Simplifying a combination involves cancelling out common factors in the numerator and denominator, often resulting in much smaller numbers, which makes the calculation manageable. This is crucial when evaluating the probability of winning a lottery prize.
Probability Calculations
Probability calculations are the backbone of determining the likelihood of various outcomes in a random event, like a lottery draw. Probability can be expressed as the ratio of favorable outcomes to the overall number of possible outcomes.

In the context of the lottery, to calculate the probability of winning, you divide the number of ways to win (the successful outcomes) by the total number of possible combinations (the total number of possibilities). This is demonstrated in the solution where the probability of winning a consolation prize is found by dividing \(\binom{56}{3} \times \binom{46}{1}\) by \(\binom{56}{5} \times \binom{46}{1}\).

Understanding the fundamental principle that probability is a fraction ranging from 0 (impossibility) to 1 (certainty) is key. For lotteries, the probabilities are typically very small because the number of possible combinations is vast in comparison to the number of winning outcomes.
Mathematical Odds
Mathematical odds are another way to represent the likelihood of an event occurring versus it not occurring. While probability measures the chance of winning, odds compare the chance of winning to the chance of losing.

In a lottery scenario, odds are calculated by taking the number of successful outcomes and setting it against the number of unsuccessful outcomes. Unlike probability, which is represented as a fraction, odds are usually represented as a ratio, like '1 to 10' or '1:10'. This means for every 1 time you win, you expect to lose 10 times.

It is important to note that odds and probability are related concepts but they are not the same. The probability of winning a lottery is usually much lower than the odds suggest, as the odds do not account for the total number of possible outcomes, only the ratio of success to failure. For instance, if the probability of winning a lottery is \(\frac{1}{1000}\), the odds are '1 to 999', signifying that for every one win, there are 999 losses.

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Most popular questions from this chapter

Are related to the SAT, described in Check Point 4 on page \(752 .\) An oil company is considering two sites on which to drill, described as follows: Site A: Profit if oil is found: \(\$ 80\) million Loss if no oil is found: \(\$ 10\) million Probability of finding oil: \(0.2\) Site B: Profit if oil is found: \(\$ 120\) million Loss if no oil is found: \(\$ 18\) million Probability of finding oil: \(0.1\) Which site has the larger expected profit? By how much?

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It is estimated that there are 27 deaths for every 10 million people who use airplanes. A company that sells flight insurance provides \(\$ 100,000\) in case of death in a plane crash. A policy can be purchased for \(\$ 1\). Calculate the expected value and thereby determine how much the insurance company can make over the long run for each policy that it sells.

You are taking a multiple-choice test that has five questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?
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