Jasper's friend Tina challenges him to a game of Unders and Overs. He's never played the game before, so Tina explains the rules:
"You bet what the total value will be when I roll two dice. Then, I roll the dice and their total determines whether you win money off me or whether I get to keep your bet. Don't worry, it's not complicated; you only have three bets that you can make.
"You can bet Under 7. If the total is under 7, you win 'even money'—the same amount as you bet. You double your money. But if the total is 7 or more, you lose your bet.
"You can bet Over 7. If the total is over 7, you win even money. It the total is 7 or less, you lose.
"You can bet 7. If the total is exactly seven, then you win three times the amount you bet —you quadruple your money! But any total except 7 means I win. So, do you want to play?"
Jasper knows that Tina hates to lose. He also knows that he can't afford to lose money right now, but he sure would like to win some from Tina. Jasper uses his powerful math skills to analyze the game and see what his chances are of coming out the winner.
Should Jasper play Unders and Overs? Is he likely to win or lose? If he stands a good chance of winning, is there a specific bet he should make each time?
How can Jasper answer these questions? Try to answer the questions yourself. Save your notes and calculations somewhere where you can refer to them again at the end of this module. You will not be graded on whether you can solve this question now, but you will be required to compare your attempt now with your second attempt at the end of the module.
Can get 7 with 4,3 or 3,4 or 5,2 or 2,5 or 6,1 or 1,6 out of 36 possibilities = 6/36
Use the same logic for above or below 7.
To determine whether Jasper should play Unders and Overs and analyze his chances of winning, he can start by calculating the probabilities of each possible outcome when rolling two dice.
There are 36 possible outcomes when rolling two dice since each die has 6 sides (6 x 6 = 36).
1. Under 7:
To calculate the probability of getting a total under 7, Jasper needs to determine the number of combinations that give a total of 2, 3, 4, 5, or 6.
- Total 2: There is only one combination (1, 1)
- Total 3: There are two combinations (1, 2) and (2, 1)
- Total 4: There are three combinations (1, 3), (2, 2), and (3, 1)
- Total 5: There are four combinations (1, 4), (2, 3), (3, 2), and (4, 1)
- Total 6: There are five combinations (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1)
The total number of combinations for totals under 7 is 1 + 2 + 3 + 4 + 5 = 15.
So, the probability of getting a total under 7 is 15/36 = 5/12.
2. Over 7:
To calculate the probability of getting a total over 7, Jasper needs to determine the number of combinations that give a total of 8, 9, 10, 11, or 12.
- Total 8: There are five combinations (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2)
- Total 9: There are four combinations (3, 6), (4, 5), (5, 4), and (6, 3)
- Total 10: There are three combinations (4, 6), (5, 5), and (6, 4)
- Total 11: There are two combinations (5, 6) and (6, 5)
- Total 12: There is only one combination (6, 6)
The total number of combinations for totals over 7 is 5 + 4 + 3 + 2 + 1 = 15.
So, the probability of getting a total over 7 is 15/36 = 5/12.
3. Exactly 7:
To calculate the probability of getting a total of exactly 7, Jasper needs to determine the number of combinations that give a total of 7.
- Total 7: There are six combinations (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1)
The total number of combinations for a total of 7 is 6.
So, the probability of getting a total of 7 is 6/36 = 1/6.
Now that Jasper knows the probabilities of each outcome, he can analyze his chances of winning:
- Probability of winning when betting Under 7: 5/12 (approximately 0.4167)
- Probability of winning when betting Over 7: 5/12 (approximately 0.4167)
- Probability of winning when betting 7: 1/6 (approximately 0.1667)
From this analysis, it is clear that Jasper's best chance of winning is when betting on 7, with a probability of 1/6. However, the probabilities for Under 7 and Over 7 are the same.
Considering that Jasper can't afford to lose money, it might be best for him to not play the game as the probabilities of winning are low overall.
Jasper should take into account his financial situation and decide accordingly whether to play or not.
To answer these questions, Jasper needs to analyze the probabilities and payouts associated with each of the three bets in the game of Unders and Overs.
Let's start by calculating the probabilities of each possible outcome when rolling two dice:
- There are 36 possible outcomes when rolling two dice (since there are 6 possible outcomes for each die, and 6 x 6 = 36 total outcomes).
- The only way to get a total of 2 is by rolling two 1s, so there is only 1 way to get a total of 2.
- Similarly, there is only 1 way to get a total of 12 by rolling two 6s.
- There are multiple ways to get each of the totals from 3 to 11. For example, there are two ways to get a total of 4 (1+3 and 3+1). We can create a chart to summarize the number of ways to get each possible total:
Total: 2 3 4 5 6 7 8 9 10 11 12
Ways: 1 2 3 4 5 6 5 4 3 2 1
Now, let's calculate the probabilities of each total:
- To calculate the probability of a specific total, we divide the number of ways to get that total by the total number of possible outcomes (36 in this case).
- For example, the probability of getting a total of 2 is 1/36 (1 way out of 36 possible outcomes).
- We can calculate the probabilities for all possible totals and summarize them in another chart:
Total: 2 3 4 5 6 7 8 9 10 11 12
Probability: 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
Next, let's calculate the payouts for each bet:
- "Under 7" bet: if the total is under 7, Jasper wins even money (the same amount he bet). If the total is 7 or above, he loses his bet.
- "Over 7" bet: if the total is over 7, Jasper wins even money. If the total is 7 or below, he loses his bet.
- "7" bet: if the total is exactly 7, Jasper wins three times the amount he bet. If the total is any other number, he loses.
Now, let's analyze each bet:
- "Under 7" bet:
- The possible outcomes that would result in Jasper winning this bet are the totals 2, 3, 4, 5, and 6.
- From the probability chart, we know the probabilities for each of these totals: 1/36, 2/36, 3/36, 4/36, and 5/36.
- The expected value of this bet can be calculated by multiplying each probability by the payout (even money) and summing them up:
- (1/36) * +1 + (2/36) * +1 + (3/36) * +1 + (4/36) * +1 + (5/36) * +1 = 15/36
- "Over 7" bet:
- The possible outcomes that would result in Jasper winning this bet are the totals 8, 9, 10, 11, and 12.
- From the probability chart, we know the probabilities for each of these totals: 5/36, 4/36, 3/36, 2/36, and 1/36.
- The expected value of this bet can be calculated by multiplying each probability by the payout (even money) and summing them up:
- (5/36) * +1 + (4/36) * +1 + (3/36) * +1 + (2/36) * +1 + (1/36) * +1 = 15/36
- "7" bet:
- The possible outcome that would result in Jasper winning this bet is the total 7.
- From the probability chart, we know the probability of getting a total of 7 is 6/36.
- The expected value of this bet can be calculated by multiplying the probability by the payout (3 times) and adding it to the expected value of losing the bet (which is the sum of probabilities for all other totals multiplied by -1):
- (6/36) * +3 + (1 - 6/36) * -1 = 6/36 + 30/36 * -1 = 6/36 - 30/36 = -24/36 = -2/3
Based on the expected values calculated for the three bets, we can conclude the following:
- The "Under 7" bet has an expected value of +15/36, which means Jasper is expected to win, on average, 15 out of 36 bets. This is the best bet in terms of expected value.
- The "Over 7" bet also has an expected value of +15/36, so it has the same winning probability as the "Under 7" bet.
- The "7" bet has an expected value of -2/3, which means Jasper is expected to lose, on average, 2 out of 3 bets.
Therefore, Jasper should play Unders and Overs and focus on making either the "Under 7" or "Over 7" bets. Both of these bets have the same winning probability, so he can choose either based on personal preference. The "7" bet is not recommended as it has a negative expected value and is more likely to result in losses.