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Expected Value for the Ohio Pick 4 Lottery In the Ohio Pick 4 lottery, you can bet \(\$ 1\) by selecting four digits, each between 0 and 9 inclusive. If the same four numbers are drawn in the same order, you win and collect \(\$ 5000\). a. How many different selections are possible? b. What is the probability of winning? c. If you win, what is your net profit? d. Find the expected value for a \(\$ 1\) bet. e. If you bet \(\$ 1\) on the pass line in the casino dice game of craps, the expected value is \(-1.4 \varphi\). Which bet is better in the sense of producing a higher expected value: a \(\$ 1\) bet in the Ohio Pick 4 lottery or a \(\$ 1\) bet on the pass line in craps?

Step by step solution

01

Calculate the Total Number of Different Selections

Each digit can be any number from 0 to 9, which gives us 10 possibilities per digit. Since there are 4 digits, the total number of different selections is defined by the formula for permutations:\[ 10^4 = 10 \times 10 \times 10 \times 10 \]Therefore, \[ 10^4 = 10000 \]So there are 10,000 different selections possible.

02

Calculate the Probability of Winning

The probability of winning is the ratio of the number of ways to win to the total number of different selections, which is \[ p = \frac{1}{10,000} = 0.0001 \]

03

Determine the Net Profit if You Win

If you win, you collect \(\$5000\) but you bet \(\$1\). Therefore, your net profit is\[ \$5000 - \$1 = \$4999 \]

04

Find the Expected Value for a \(\$1\) Bet

The expected value (EV) is calculated by the formula\[ EV = (\text{Probability of Winning} \times \text{Net Profit}) + (\text{Probability of Losing} \times \text{Loss}) \]Substituting the known values:\[ EV = (0.0001 \times 4999) + (0.9999 \times -1) \]\[ EV = 0.4999 - 0.9999 = -0.5 \]Thus, the expected value for a \(\$1\) bet in the Ohio Pick 4 lottery is \(-0.5\).

05

Compare with Casino Craps Bet

The expected value of a \(\$1\) bet on the pass line in the casino dice game of craps is \(-1.4\). Comparing the two expected values, since\[ -0.5 > -1.4 \]the expected value for the Ohio Pick 4 lottery bet is higher, making it a better bet in this sense.

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Key Concepts

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

Probability Calculation

Probability is a measure of how likely an event is to occur. In the Ohio Pick 4 lottery, each digit in your selection can range from 0 to 9, providing 10 possibilities per digit. Since you are selecting 4 digits, there are a total of \(10^4 \) or 10,000 possible combinations. To win the lottery, your chosen combination must match the winning numbers exactly, so the probability of winning is just one out of 10,000: \(\displaystyle p = \frac{1}{10,000} = 0.0001 \). This means there's a very slim chance, only 0.01%, to win.

Net Profit Calculation

Net profit is the amount of money you gain after subtracting any costs or losses. When you win the Ohio Pick 4 lottery, you collect \(5000. However, since you originally bet \)1, your net profit is calculated by subtracting your initial bet from your winnings: \ \$5000 - \$1 = \$4999 \. This means the net profit is \$4999 when a winning combination is achieved.

Permutations in Lottery

Permutations are different ways of arranging a set of items. In lottery mathematics, the order in which numbers are arranged matters. For the Ohio Pick 4 lottery, each selection of four digits is unique, so we're dealing with permutations. With each digit having 10 options, the total number of permutations is given by \(10^4 = 10 \times 10 \times 10 \times 10 = 10,000 \). This means there are 10,000 unique ways to arrange four digits, making it clear why there's only a 1 in 10,000 chance to win.

Expected Value in Lottery Mathematics

Expected value (EV) is a concept that helps determine the average outcome of a random event. For a \$1 bet in the Ohio Pick 4 lottery, we calculate EV using: \( EV = ( \text{Probability of Winning} \times \text{Net Profit}) + (\text{Probability of Losing} \times \text{Loss} ) \). Here, the EV calculation becomes:\ EV = (0.0001 \times 4999) + (0.9999 \times -1) = 0.4999 - 0.9999 = -0.5 \.
The negative result shows that, on average, you can expect to lose \(0.50 for each \)1 bet. Comparing this to the \$1 bet on the pass line in craps, which has an EV of -1.4, reveals that the Ohio Pick 4 bet has a higher expected value, making it a statistically better bet.