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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblem 5.1.29d
Chapter 5, Problem 5.1.29d

Expected Value for the Florida Pick 3 Lottery In the Florida Pick 3 lottery, you can bet \$1 by selecting three digits, each between 0 and 9 inclusive. If the same three numbers are drawn in the same order, you win and collect \(500.


d. Find the expected value for a \)1 bet.

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Step 1: Understand the problem. The expected value (E[X]) is a measure of the average outcome of a random event over the long run. It is calculated as the sum of all possible outcomes, each weighted by its probability of occurrence. Here, the outcomes are winning \$500 or losing \$1.
Step 2: Calculate the probability of winning. To win, the three digits you select must match the three digits drawn in the exact order. Since each digit can be any number from 0 to 9, there are 10 × 10 × 10 = 1000 possible combinations. Therefore, the probability of winning is 1/1000.
Step 3: Calculate the probability of losing. If the probability of winning is 1/1000, then the probability of losing is the complement, which is 1 - 1/1000 = 999/1000.
Step 4: Determine the monetary outcomes. If you win, you gain \$500. If you lose, you lose your \(1 bet, which is equivalent to a monetary outcome of -\)1.
Step 5: Compute the expected value. Use the formula E[X] = (P(win) × Outcome(win)) + (P(lose) × Outcome(lose)). Substitute the probabilities and outcomes into the formula: E[X] = (1/1000 × 500) + (999/1000 × -1). Simplify the expression to find the expected value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Expected Value

Expected value is a fundamental concept in probability and statistics that represents the average outcome of a random event when repeated many times. It is calculated by multiplying each possible outcome by its probability and summing these products. In the context of gambling, it helps determine whether a bet is favorable or not.
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Probability

Probability quantifies the likelihood of a specific event occurring, expressed as a number between 0 and 1. In the Florida Pick 3 lottery, the probability of winning by selecting the correct three-digit combination in the correct order is calculated based on the total number of possible combinations, which is 1,000 (from 000 to 999). Understanding probability is crucial for calculating expected value.
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Payoff

Payoff refers to the amount of money won from a bet if the outcome is favorable. In the Florida Pick 3 lottery, the payoff for a winning $1 bet is $500. The expected value calculation incorporates both the payoff and the probability of winning to assess the overall value of the bet, helping players make informed decisions.
Related Practice
Textbook Question

In Exercises 25–28, find the probabilities and answer the questions.



Too Young to Tat Based on a Harris poll, among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. Assume that five adults who regret getting tattoos are randomly selected, and find the indicated probability.


d. If we randomly select five adults, is 1 a significantly low number who say that they were too young to get tattoos?

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Textbook Question

Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay \(1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect \)5000.


e. If you bet \$1 in North Carolina’s Pick 3 game, the expected value is Which bet is better in the sense of a producing a higher expected value: A \$1 bet in the North Carolina Pick 4 game or a \$1 bet in the North Carolina Pick 3 game?

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Textbook Question

Using Probabilities for Significant Events


d. Is 1 a significantly low number of matches? Why or why not?

Textbook Question

Using Probabilities for Significant Events


c. Which probability is relevant for determining whether 1 is a significantly low number of matches: the result from part (a) or part (b)?


Textbook Question

Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay \(1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect \)5000.


d. Find the expected value.

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Textbook Question

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.

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d. Which probability is relevant for determining whether 40 first lines for Democrats is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 40 first lines for Democrats significantly high?


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