Math for the People

Subsection 6.6.1 Introduction to Probability

The probability of a particular event occurring is a number between 0 and 1 (or, equivalently, between 0% and 100%) which measures how likely that event is to occur. If something has a probability of 0, that means it will never happen. If it has a probability of 1, then it will always happen, no matter what.

Usually, when we study probability, we talk about simple situations where it's easy to determine the probability. For example, if you roll a 6-sided die, the probability that you get a 6 is \(\frac{1}{6}\) because you're looking at 1 possibility out of 6 total.

However, in the real world, probabilities are usually much more complicated to determine. To calculate your credit score, credit reporting agencies use many different factors - how long you've had loans or credit cards, how many times you have been late paying your bills, and the types of loans and credit cards that you have. There is no simple formula to calculate how likely you are to repay a loan. The credit score summarizes the complicated calculations that have gone into determining the likelihood of you paying back money that you borrow.

Although it is an oversimplification of the actual situation, we'll use the credit score as a simple measure of probability so that we can see an example of how it works. We'll treat a score of 300 - the lowest score - as meaning that the probability you pay back a loan is 0%. We'll say that a credit score of 850 means that there is a 100% chance that you pay back the loan on time. We'll treat every score between as a number between 0 and 100% using the formula

So a credit score of 700, in our simplification would mean a \(\frac{700-300}{850-300} = 0.73 = 73\%\) chance of repaying the loan on time.

Subsection 6.6.2 Expected Value

Once banks determine how likely you are to repay a loan, they can make a decision about whether they will loan you money, whether that's as a student loan, credit card, or something else. They use the credit score to determine the amount of money they can expect to make off of the loan. If you don't repay the loan, they will lose all of the money that they loaned you. If you do repay the loan, then they make money - in the form of the interest that you pay back on the loan. This is called the expected value of the loan. It depends on both the probability that a borrower will pay back the loan, the amount of money being borrowed, and the amount of interest that the bank will make off of the loan.

Before we look at the expected value of a loan, let's define the expected value in general. Expected value works by using the probability to weight the reward of different events happening. Consider this example: Say a friend makes you an offer. They'll flip a coin. If the coin comes up heads, they'll pay you $1. If the coin comes up tails, you pay them $1.

If you play the game just once, you'll end up either $1 richer or $1 poorer - there's a 50% chance of either. If you play it twice, you either win both, win the first and lose the second, lose the first and win the second, or lose both - so there are 4 possibilities. If you win both times, you're $2 richer. If you win one and lose one (either of the middle two situations) you're back where you started - $0 richer. If you lose both, you're $2 poorer. If you average out the different scenarios, you end up exactly where you started - there's exactly the same chance of winning money as losing. So we say the expected value is $0 - on average, we don't expect to end up with any more money than we started with.

Expected value depends on probability - so it certainly isn't saying that if you play that game with your friend, you won't win any money! You could play 10 times and win every time...it just isn't very likely (the chance of that happening is a little less than 1/1000)

Let's look at an example of how this works, using insurance as an example. When an insurance company sells you insurance, they calculate the probability that you will use it - and how much they'll have to pay you if you do use it. If you don't use the insurance, they "win" the amount of money that you pay for the insurance - if you do use it, they lose whatever they have to pay out for the claim (minus whatever you paid them). On average, over lots and lots of customers, they'll earn the expected value from each customer. Let's look at an example.

An insurance company calculates that there is a 1% chance that a driver has a car accident each year, and that on average they will have to pay $15,000 per accident. They charge $600 a year for insurance. How much can they expect to earn, on average, from each customer?

The probability of having an accident is 1% (0.01), which means that the probability of not having an accident is 99% (0.99). If you have an accident, the insurance company loses $15,000, minus the $600 you paid them, for a net "reward" of \(\$600 - \$15,000 = -\$14,400\text{.}\) If you don't have an accident, they earn a "reward" of $600 - the amount you paid for your insurance. The expected value, then is:

On average, they can expect to earn $450 per customer that they sell insurance to.

Give it a try yourself - here are some examples to work out.

Let's look at example explicitly about how payday lenders operate. Most payday lenders make their money because people renew their loans week after week, paying larger fees to borrow the money for a longer period of time.

Subsection 6.6.3 How Banks Make Lending Decisions

Banks use the idea of risk to make decisions about how much money they will lend. We'll look at a simplified example of how this works, though the reality is, of course, more complicated. This will still give you an idea, fundamentally of how it works.

Say that you want to borrow $1000 from a bank via an unsecured loan. If the loan were secured, and you didn't pay it back, the bank could seize your house or car or whatever you had secured the loan with. Since it's unsecured, the bank will lose all of the money that they loaned you if you don't pay it back. Therefore, they want to make the expected value of loaning you money positive (for them) - that is, they want to ensure that on average the money that you pay them in interest makes up for the risk of you not paying the loan back.

Let's look at the situation where the bank loans you $1000 for 1 year, simple interest, and let's assume that they decide there's a 10% chance you don't pay the loan off. If the interest rate is r, the expected value calculations look like this:

Let's do a more complicated example, with compound interest.