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

## Step by step solution

## Understand the Problem

## Calculate the actual odds against winning

## Find the payoff odds

## Verify the given odds

## Key Concepts

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

###### Probability

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

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).

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

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: \(\text{\textdollar} 1\)

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 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 \(\text{\textdollar} 1\).

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.