Money and You: Course Notes

One common coincidence that people experience is running into someone with the same birthday. Let’s briefly exam how common or uncommon this is.

Suppose there are a number of people in a room. What is the probability that at least two people share a birthday?

Notice that this question is difference than, “what is the probability that someone has your birthday?” We are just asking if there are some people in the group that share a birthday. It turns our that this question is more easily answered by asking the opposite question. Instead we ask, “what is the probability that no one shares a birthday with anyone else?” The reason why is it is easier to examine this question mathematically.

We won’t go over the mathematics in full, but we’ll explore a bit. Let’s suppose that 3 people come into a room one-by-one. Let’s also suppose (for simplicity) that February 29th birthdays aren’t a thing and that birthdays are evenly distributed over a whole year. The first person walks into the room followed by the second. What is the probability that person number 2 does not have the same birthday as person 1? It is \(\frac{364}{365}\approx 99.73\%\text{.}\) Now the third person walks in. What is the probability that that person does not share a birthday with either person 1 or person 2? It would be \(\frac{363}{365}\approx 99.45\%\text{.}\) (Because there are 363 other days in the year that are not the birthdays of persons 1 or 2.)

Now here is the important bit. What is the probability that all three have different birthdays? We calculate it as a process. First, person 2 has a different birthday than person 1. Then, person 3 has a different birthday than person 2. The probability is calculated as the product of those two events: \(\frac{364}{365}\times \frac{363}{365}\approx 99.12\%\text{.}\) So, it’s very likely that the three people have different birthdays.

Now what if we increase the number of people? If we add a fourth person, the probability that none of the four people share a birthday with someone else in the group is: \(\frac{364}{365}\times \frac{363}{365}\times\frac{362}{365}\approx 98.36\%\text{.}\) So, if there is a group of four people, there is a \(100\% - 98.36\% = 1.64\%\) chance that at least two people share a birthday.

We can keep going:

5 people in a group: 2.71%chance of a shared birthday.

6 people in a group: 4.05%chance of a shared birthday.

7 people in a group: 5.62%chance of a shared birthday.

8 people in a group: 7.43%chance of a shared birthday.

Notice that the chances of a shared birthday is growing with the number of people. Also notice that the chance is growing faster and faster. This is the important observation.

Now, if you have 23 people in the group, if you do the math, you’ll get that there is a 50.73%chance that there is a shared birthday in a group. That means that when there are 23 people in a group, it is actually more likely that at least two people share a birthday than it is that everyone has different birthdays! (This does not means that it is more likely that someone has your birthday than not. It means that it is more likely that there are some people who share a birthday.) That mathematical fact is not obvious, bit it is important. If you were sitting in a class with at least 23 students, do you intuitively believe that it is likely that some people in the room share a birthday? Probably not, but it is true.

Here is the important takeaway from the birthday problem. It is indeed unlikely that when two people meet that they have the same birthday. However, when there is a group of people, there are a lot of “chances” for there to be a shared birthday. With each person, the likelihood that no one has a shared birthday gets smaller and smaller, making the likelihood of some shared birthday higher and higher.

Let’s consider another type of problem. Suppose the WSU SGA is holding an event in which there are three games to play. For each game, only 20%of people win. How likely is it that you win at least one of the games?

Again, it is mathematically easier to find the probability that you do not win any of the games. For each game, there is an 80%chance that you’ll lose. The probability that you lose all three would be \(0.8 \time 0.8 \time 0.8 = 0.512\) or 51.2%. This means there is a 48.8%chance that you’ll win at least once. Were you expecting that probability to be that high?

What we are doing here is not looking at one event happening on its own. Instead, we are looking at all possible events at once and asking what is the likelihood that at least one of the possible things happen. Or rather, what is the likelihood that none of the events happen? Or even, what is the likelihood that no coincidences happen? The more possibilities for some coincidence to happen, the more likely that something will happen.

Here is some good news. You probably will not die from heart disease. About 20.1%of Americans die from heart disease, which means 79.9%do not. You probably won’t die from cancer. About 19%of Americans will die from cancer. So, about 81%of Americans do not die from cancer. You probably won’t die from an unintentional injury like falling or getting in a car accident. About 7%of Americans die from accidents, which means 93%of Americans do not die from accidents. You probably won’t die from a stroke. About 5.3%of American die from a stroke, meaning 94.7%of Americans do not.

Here is the bad news. If we ask, what is the probability that you won’t die from heart disease, cancer, an accident, or a stroke, the probability because \(0.799 \times 0.81 \time 0.93 \time 0.947 \approx 0.57\) or 57%. This means there is a 43%chance that you will die from one of those causes, which is very high. To make matters worse, if you are wondering what the probability is of simply passing on in your sleep without disease or disfunction, it’s really, really small. (Please note that there are a lot of other factors that contribute to these proababilities. So, these figures likely do not apply exactly to you individually.)

Here is the point to take away. Individually, the likelihood of one particular thing happening may be small. However, the probability that at least one bad thing happening gets quite large.

It is all about risk exposure. Take today as an example. Today, it is very unlikely that you’ll experience something that will require you to get emergency medical care (poisoning, injury, disease, etc.) Tomorrow, it will be just as unlikely. But what about some day in the next year? The likelihood that you will need emergency medical care at some point in the next year is substantially higher than the likelihood that you will need it on one particular day.