Math Kids

Home Math Kids Science Kids

Risk: is it worth it?

1. problem You are thinking of cheating on your math test.
If you cheat, and don't get caught, you will get a better grade. If you cheat, and get caught, you will fail the exam and get in trouble. Putting aside the moral and ethical issues, how do you decide if you should cheat or not?

2. intuition
Your decision is based on what you expect to gain by cheating balanced against the chance of getting caught and the penalty for getting caught.
Let's say that if you don't cheat, you expect to score 60/100 on your exam. If you do cheat you expect to score 100/100. We need to factor in one more thing. If you cheat and get caught, you will score 0/100 (let's ignore, for now, the other penalties associated with getting caught cheating).

Naturally, you will decide to cheat if doing so will increase your exam score. But since you don't know beforehand if you will get caught or not, you don't know if cheating will yield a score of 100 or 0.

3. math
Let's say that the chance of getting caught is 50%. This means that the probability of getting caught is 50/100 or 1/2. The way to think about probabilities is to imagine taking, and cheating on, a large number of exams. After taking all of these exams, you can expect to get caught 1/2 of the time. On half of the exams you will score 100 and on the other half you will score 0. So your average score is 50 (100 x 1/2 + 0 x 1/2). This means that on any single exam, your expected score, if you cheat, is 50. This is the score that we compare against the score you receive for not cheating, 60. In this case, cheating won't pay off.

Perhaps it seems strange to say that you expect to score 50 on the exam, since you will never actually receive a score of 50 (you will score 100 or 0). This expected score arises because of the uncertainty in getting caught -- it allows us to combine the possible scores with their probabilities so that we have a single number upon which we can make a decision.
Let's now say that the chance of getting caught goes down to 25%. In this case you will get caught 25/100 = 1/4 times, and not get caught 75% = 75/100 = 3/4 times. Your expected score now is 75 (100 x 3/4 + 0 x 1/4). In this case, cheating does pay off -- you can expect to gain an extra 15 points over not cheating.
Now, let's factor in the other penalties for cheating. You will get in trouble with your teacher, principal and your parents. You might get detention, suspended, lose TV/video game privileges, etc. These should be factored into your decision. Let's say that the punishment is the equivalent of 200 exam points. Now, the score you receive for cheating and getting caught is -200 (instead of just 0). Your expected score now becomes 25 (100 x 3/4 - 200 x 1/4). In this case, cheating does not pay off because it is less than the score of 60 that you would receive by not cheating.

4. summary
When faced with a decision where the outcome of one or more of your choices is uncertain, you can make a mathematically sound decision by combining the cost and probability associated with each option. The expected cost of any option is simply the probability times the cost.
This strategy applies to many real-world situations:

You can buy your favorite candy bar (A) for $2, or two of a new candy bar (B) for $1 each. Which should you buy? Let's assign a score of 100 to the pleasure of eating candy bar A. You have never tried candy bar B before, so you don't know if it will bring you more pleasure or not. Let's say that the probability of you liking candy bar B as much as A is 25% = 25/100 = 1/4. Then, the expected pleasure of candy bar B is 25 (100 x 1/4), and the expected pleasure of both of them is 50 (100 x 1/4 + 100 x 1/4). In this case, you should buy candy bar A -- quality wins over quantity. If however, the price of candy bar B was just $0.25, then you could buy eight of them for $2, and your expected pleasure would be 200 (8 x 100 x 1/4), and you should try the new candy bar -- quantity wins over quality.

A lottery ticket costs $1 and the jackpot is $1,000,000. Should you buy the ticket? We need to compare the cost of the ticket ($1) with the expected payoff of buying the ticket. Let's say that the chance of winning is 1/50,000,000 (this is roughly the chance of winning the New York State lottery). The expected payoff is 2 cents (1,000,000 x 1/50,000,000). In this case, you should not buy a ticket.

Your football team just scored a touchdown. You can kick a field-goal to add 1 point to your score, or try for 2 more points by running/passing the ball into the end-zone. Which should you do? Let's say that the probability of making the field-goal is 90% = 90/100 = 9/10, and that the probability of making the 2-point conversion is 25% = 25/100 = 1/4. Then, the expected payoff of the field goal is 0.9 points (1 x 9/10) and the expected payoff of the 2-point conversion is 0.5 (2 x 1/4). In this case, you should kick the field-goal (assuming that you are not down by 2 points with 10 seconds left on the clock).

Home Math Kids Science Kids

Risk: is it worth it?
1. problem	You are thinking of cheating on your math test. If you cheat, and don't get caught, you will get a better grade. If you cheat, and get caught, you will fail the exam and get in trouble. Putting aside the moral and ethical issues, how do you decide if you should cheat or not?
2. intuition	Your decision is based on what you expect to gain by cheating balanced against the chance of getting caught and the penalty for getting caught. Let's say that if you don't cheat, you expect to score 60/100 on your exam. If you do cheat you expect to score 100/100. We need to factor in one more thing. If you cheat and get caught, you will score 0/100 (let's ignore, for now, the other penalties associated with getting caught cheating). Naturally, you will decide to cheat if doing so will increase your exam score. But since you don't know beforehand if you will get caught or not, you don't know if cheating will yield a score of 100 or 0.
3. math	Let's say that the chance of getting caught is 50%. This means that the probability of getting caught is 50/100 or 1/2. The way to think about probabilities is to imagine taking, and cheating on, a large number of exams. After taking all of these exams, you can expect to get caught 1/2 of the time. On half of the exams you will score 100 and on the other half you will score 0. So your average score is 50 (100 x 1/2 + 0 x 1/2). This means that on any single exam, your expected score, if you cheat, is 50. This is the score that we compare against the score you receive for not cheating, 60. In this case, cheating won't pay off. Perhaps it seems strange to say that you expect to score 50 on the exam, since you will never actually receive a score of 50 (you will score 100 or 0). This expected score arises because of the uncertainty in getting caught -- it allows us to combine the possible scores with their probabilities so that we have a single number upon which we can make a decision. Let's now say that the chance of getting caught goes down to 25%. In this case you will get caught 25/100 = 1/4 times, and not get caught 75% = 75/100 = 3/4 times. Your expected score now is 75 (100 x 3/4 + 0 x 1/4). In this case, cheating does pay off -- you can expect to gain an extra 15 points over not cheating. Now, let's factor in the other penalties for cheating. You will get in trouble with your teacher, principal and your parents. You might get detention, suspended, lose TV/video game privileges, etc. These should be factored into your decision. Let's say that the punishment is the equivalent of 200 exam points. Now, the score you receive for cheating and getting caught is -200 (instead of just 0). Your expected score now becomes 25 (100 x 3/4 - 200 x 1/4). In this case, cheating does not pay off because it is less than the score of 60 that you would receive by not cheating.
4. summary	When faced with a decision where the outcome of one or more of your choices is uncertain, you can make a mathematically sound decision by combining the cost and probability associated with each option. The expected cost of any option is simply the probability times the cost. This strategy applies to many real-world situations: You can buy your favorite candy bar (A) for $2, or two of a new candy bar (B) for $1 each. Which should you buy? Let's assign a score of 100 to the pleasure of eating candy bar A. You have never tried candy bar B before, so you don't know if it will bring you more pleasure or not. Let's say that the probability of you liking candy bar B as much as A is 25% = 25/100 = 1/4. Then, the expected pleasure of candy bar B is 25 (100 x 1/4), and the expected pleasure of both of them is 50 (100 x 1/4 + 100 x 1/4). In this case, you should buy candy bar A -- quality wins over quantity. If however, the price of candy bar B was just $0.25, then you could buy eight of them for $2, and your expected pleasure would be 200 (8 x 100 x 1/4), and you should try the new candy bar -- quantity wins over quality. A lottery ticket costs $1 and the jackpot is $1,000,000. Should you buy the ticket? We need to compare the cost of the ticket ($1) with the expected payoff of buying the ticket. Let's say that the chance of winning is 1/50,000,000 (this is roughly the chance of winning the New York State lottery). The expected payoff is 2 cents (1,000,000 x 1/50,000,000). In this case, you should not buy a ticket. Your football team just scored a touchdown. You can kick a field-goal to add 1 point to your score, or try for 2 more points by running/passing the ball into the end-zone. Which should you do? Let's say that the probability of making the field-goal is 90% = 90/100 = 9/10, and that the probability of making the 2-point conversion is 25% = 25/100 = 1/4. Then, the expected payoff of the field goal is 0.9 points (1 x 9/10) and the expected payoff of the 2-point conversion is 0.5 (2 x 1/4). In this case, you should kick the field-goal (assuming that you are not down by 2 points with 10 seconds left on the clock).