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Chapter 5: Problem 59

In the Illinois Lottery game Little Lotto, an urn contains balls numbered 1 to \(30 .\) From this urn, 5 balls are chosen randomly, without replacement. For a \(\$ 1\) bet, a player chooses one set of five numbers. To win, all five numbers must match those chosen from the urn. The order in which the balls are selected does not matter. What is the probability of winning Little Lotto with one ticket?

Short Answer

Expert verified
The probability of winning Little Lotto with one ticket is \( \frac{1}{142506} \).

Step by step solution

01

Understand the Problem

We need to determine the probability of winning Little Lotto by matching all five selected numbers from an urn containing 30 balls.
02

Define Total Outcomes

Since we are choosing 5 balls out of 30 without regard to order, we need to calculate the number of combinations of 5 balls from 30. This can be done using the combination formula \[ C(n, k) = \frac{n!}{k!(n-k)!} \], where \( n = 30 \) and \( k = 5 \).
03

Calculate Total Combinations

Using the combination formula, we get \[ C(30, 5) = \frac{30!}{5!(30-5)!} = \frac{30!}{5! \cdot 25!} \]. This simplifies to:\[ \frac{30 \times 29 \times 28 \times 27 \times 26}{5 \times 4 \times 3 \times 2 \times 1} = 142506. \] So, there are 142,506 possible combinations.
04

Define Successful Outcomes

There is only one successful outcome, which is the set of 5 numbers that the player has chosen.
05

Calculate Probability

The probability is calculated by dividing the number of successful outcomes by the total number of possible outcomes: \[ P(\text{winning}) = \frac{1}{C(30, 5)} = \frac{1}{142506} \].

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Key Concepts

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

Combinatorics
Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. It plays a key role in probability and lottery games. In combinatorics, understanding how objects can be combined or arranged helps to solve problems involving finite sets. For instance, when determining how many ways 5 balls can be chosen from 30, we use combinatorial methods. By applying formulas and principles from combinatorics, such problems become more manageable and systematic.
Probability
Probability measures how likely an event is to occur. It ranges between 0 and 1, where 0 means the event is impossible and 1 indicates certainty. In lottery games, probability helps to understand the chances of winning. Calculating probability involves dividing the number of successful outcomes by the total number of possible outcomes. For example, in the Little Lotto, we calculate the probability of winning by considering only one favorable outcome (the player's chosen numbers) against all possible combinations of numbers drawn from the urn.
Lottery Games
Lottery games are forms of gambling where winners are selected through a random drawing. These games often involve selecting numbers from a larger set and rely heavily on probability and combinatorics. To win, the selected numbers must match the drawn numbers. There are various types of lottery games worldwide, differing in their rules and odds. Understanding the mathematical underpinnings can give deeper insights into how these games operate and the likelihood of winning.
Combinations Formula
The combinations formula is used in combinatorics to determine the number of ways a subset of items can be selected from a larger set, without regard to order. The formula is given by \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, \( k \) is the number of items to choose, and \( ! \) denotes factorial. In the given exercise, \( C(30, 5) = \frac{30!}{5!(30-5)!} = 142506 \). This tells us there are 142,506 ways to pick 5 numbers from 30. The combinations formula simplifies complex problems, helping in calculating the probability of winning scenarios like the lottery.

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Most popular questions from this chapter

This past semester, I had a small business calculus section. The students in the class were Mike, Neta, Jinita, Kristin, and Dave. Suppose I randomly select two people to go to the board to work problems. What is the probability that Dave is the first person chosen to go to the board and Neta is the second?

Conduct a survey in your school by randomly asking 50 students whether they drive to school. Based on the results of the survey, approximate the probability that a randomly selected student drives to school.

How many different 10 -letter words (real or imaginary) can be formed from the letters in the word STATISTICS?

A gene is composed of two alleles. An allele can be either dominant or recessive. Suppose a husband and wife, who are both carriers of the sickle- cell anemia allele but do not have the disease, decide to have a child. Because both parents are carriers of the disease, each has one dominant normal-cell allele and one recessive sickle-cell allele. Therefore, the genotype of each parent is \(S s .\) Each parent contributes one allele to his or her offspring, with each allele being equally likely. (a) List the possible genotypes of their offspring. (b) What is the probability that the offspring will have sickle-cell anemia? In other words, what is the probability the offspring will have genotype \(s s ?\) Interpret this probability. (c) What is the probability that the offspring will not have sickle-cell anemia but will be a carrier? In other words, what is the probability that the offspring will have one dominant normal-cell allele and one recessive sickle- cell allele? Interpret this probability.

Find the value of each combination. $$_{10} C_{2}$$
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