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Chapter 4: Problem 41

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?

Short Answer

Expert verified
The actual odds against winning are 9999:1, the payoff odds are 4999:1, and the website accurately states the odds as 1:10,000.

Step by step solution

01

Understand the Problem

In this exercise, calculate the odds related to the Kentucky Pick 4 lottery. Given: Probability of winning = \frac{1}{10,000}, payout for winning = \(5000, net profit for winning = \)4999.
02

Calculate the actual odds against winning

The actual odds against winning are calculated as the ratio of losing outcomes to winning outcomes. There are 9999 losing outcomes and 1 winning outcome. Thus, the odds against winning are 9999 to 1.
03

Find the payoff odds

Payoff odds are calculated as (net profit)/(cost of the bet). Here, the net profit is \(4999, and the cost of the bet is \)1. Therefore, the payoff odds are 4999 to 1.
04

Verify the given odds

According to the website, the odds of winning are stated as 1:10,000. This is accurate since the probability of winning indeed indicates that for every 10,000 tickets, one is likely to win.

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Key Concepts

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

Probability
Probability is a measure of how likely an event is to occur. In the context of the Kentucky Pick 4 lottery, probability is the chance of picking the right combination of four digits in the exact order.
The probability of winning can be calculated using the formula:
\(\text{Probability of Winning} = \frac{1}{\text{Total Number of Outcomes}}\).
Here, the total number of possible outcomes is 10,000 because each of the four digits can be anything between 0 and 9 (10 choices per digit).
Thus, the probability of winning is \(\frac{1}{10,000}\) or 0.0001.
This means that out of 10,000 tickets, only one is expected to win. Therefore, it is very unlikely to win.
Odds
Odds represent the ratio of favorable outcomes to unfavorable outcomes. In lotteries, odds are often given to help participants understand their chances of winning.
For the Kentucky Pick 4, the odds against winning are calculated by comparing the number of losing outcomes against the one winning outcome.
  • Since the probability of winning is \(\frac{1}{10,000}\), the number of losing outcomes is 9999 (10,000 - 1).
The actual odds against winning are hence 9999 to 1.
This means that for every 10,000 attempts, 9999 will result in a loss and only 1 will result in a win. Understanding this ratio helps you grasp the risk involved.
Net Profit
Net profit is the gain after subtracting the cost of an investment. In lottery terms, it is how much you win minus how much you spent to enter.
For a straight bet in the Kentucky Pick 4 lottery, we need to consider:
  • The payout for winning: \(\text{\textdollar} 5000\)
  • Cost of the bet: \(\text{\textdollar} 1\)
Therefore, the net profit if you win is \(\text{\textdollar} 5000 - \text{\textdollar} 1 = \text{\textdollar} 4999\).
This calculation is practical for understanding the effective earnings you'll have after subtracting the amount you paid for the ticket.
Payoff Odds
Payoff odds indicate how much you can win compared to the cost of your bet. They are crucial in evaluating the attractiveness of betting games.
Payoff odds are calculated using the formula:
\(\text{Payoff Odds} = \frac{\text{Net Profit}}{\text{Cost of the Bet}}\).
  • Here, the net profit is \(\text{\textdollar} 4999\)
  • Cost of the bet is \(\text{\textdollar} 1\).
The payoff odds would be \(4999 : 1\).
This means that for every dollar you bet, you stand to gain 4999 dollars if you win. Understanding payoff odds can guide you in making more informed gambling decisions.

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Most popular questions from this chapter

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