Above: Image © Gearstd, iStockPhoto.com
Probabilities are fun! That’s right, you read that correctly. Not only are they fun, but once you understand them you can make people think you have amazing abilities!
Imagine this. You’re at a party, but the night is young and things haven’t quite picked up yet. You want to have some fun, so you make a friendly bet with the someone sitting beside you.
You say, “I bet that among the 30 people in the room, there will be at least two people who have birthdays on the exact same date.”
“No way!” your friend says. “There are 365 days in a year. I don’t believe you. The bet is on!”
What do you think your probability of winning are? Hint: they’re quite good!
Before we get into how this works, allow me to formally introduce the concept of probability.
A probability is a statement about how likely it is that something specific will happen in the future. To find probability, you divide the number of times something will happen by the number different possible outcomes.
Imagine, for example, that you flip a coin. What is the likelihood of that coin toss coming out heads? A coin toss has two possible outcomes: heads or tails. So, the chance of getting the specific outcome heads is 1 divided by 2!
Probabilities are usually expressed in one of three ways:
- As a fraction: The probability of getting heads is 1/2
- As a decimal value ranging from 0 to 1: The probability of getting heads is 0.5
- As a percentage: The probability of getting heads is 50%
All three representations say the same thing: the probability of you getting heads on that coin flip.
Note that if you sum up the probabilities of all possible outcomes, you get 1, or 100%. That’s a sure thing! But you can never have a probability greater than 1. Think of it this way: on a given coin toss, you are 100% sure to get either heads or tails. There are just no other outcomes possible.
Now, imagine that you flip the coin twice. There are four possible ways this scenario can go:
There are a bunch of ways that you can calculate the probability of getting heads on both of those flips. The most straightforward way is to list all of the possible outcomes (like we just did) and count the number of times you get heads on both flips. If we do this, we see that the probability is 1/4 (or 25%). That’s because there is only one case out of 4 in which both coin flips are heads.
But you can also come to this answer by doing some math. Getting heads on two flips is actually two separate events:
- getting heads on the first flip
- getting heads on the second flip.
If you multiply the probabilities of two events, you get the probability of both events happening. So the probability of getting heads on the first flip is 1/2, and the probability of getting heads on the second flip is 1/2. Thus the probability of getting heads on both flips will be:
1/2 x 1/2 = 1/4
Finally, you can calculate the likelihood of something happening by subtracting the probability of it not happening from 1. The probability of not getting two heads in a row is 3/4; if you take a look at the list we made before you’ll see that out of the 4 possible outcomes, three of them do not contain two heads. Thus, the probability of getting two heads in a row is 1 - 3/4 or 1/4.
Back to the party
Let’s now return to the party, and the bet you’ve made. To keep things simple, let’s subtract the probability that you lose the bet from 1. This will give you your probability of winning.
Start by selecting a random person in the room. Ask when their birthday is, or look it up online. Write it down on a napkin. In order for you to end up losing the bet, the second person you select at random should have a birthday on any one of the 364 remaining days of the year. So far, you have a 364/365 chance of losing, leaving you with a 1/365 chance of winning – your chances of losing, subtracted from 1.
You continue by selecting a third person in the room. If the first two people had a different birthday date, in order for you to end up losing, this third person would have to be born on any one of the other 363 days of the year. That reduces your chances of losing to:
364/365 x 363/365
If you carry out the operations a couple of more times, you will quickly notice a pattern. With each new person, you restrict the possible dates on which the following person can be born so that none of the dates seen so far overlap. This way, your chances of losing are reduced faster and faster. Meanwhile, your chances of winning increase continuously!
Continue this for the remaining 27 people. If each time your probability of losing drops and your probability of winning goes up, then by person #27, you’ll have a 29% chance of losing and an impressive 71% chance of winning! Now, those are some probabilities I like!
Did you know? Many people use the word “odds” when they mean “probability.” Both can be used to determine the likelihood of something, but they are calculated in different ways. A probability is an event happening divided by all other outcomes. Odds are the ratio of the number of ways the event can happen to the number of ways it can’t.
Using the coin example from earlier, your odds of getting heads are 1:1. There is 1 way you can get heads, and 1 way you can’t (that is, if you get tails).
Meanwhile, your odds of getting heads twice in a row are 1:3. That 3 represents the 3 ways you could get a different result (tails twice in a row, heads followed by tails, tails followed by heads).
If you understand them, probabilities can help you make choices directed by more than plain blind luck. They can help you see the world more clearly, and understand why some events happen the way they do. If you look carefully, you will find probabilities all around you! So, I encourage you to arm yourself with this knowledge and go looking for them. Happy hunting!
Did you know? If you flip a coin five times and get heads each time, you might feel like the chance of getting heads on that sixth flip is very small. This is called the gambler’s fallacy. The truth is, your chance of getting heads is still 1/2.
Let’s talk about it
- Can you think of some other areas in the news, or in your life, where you might hear probabilities?
- Explain, in your own words, the probabilities in the following examples:
- A medical test correctly diagnoses a disease 88.5% of the time.
- The probability of precipitation tonight is 60%.
- Would you worry about these probabilities? Why or why not?
- If a baseball player has a batting average of 300, would you want him on your team? Why or why not? (Tip: if you don’t watch or play baseball, you may need to do a bit of extra research to answer this question.)
The gambler's fallacy explained? Misguided belief in the big win just around the corner could be down to brain activity (2014)
Connor, The Independent
Probability and the Birthday Paradox (2012)
Daytona State College Academic Support Centre