A good non-technical definition of probability is hard to come by, so we’ll go with “the chance that an event will occur.” Mathematicians will use set theory and calculus, but we’ll keep things simple today. But a warning, there will be fractions!
A sense of probability is innate in all of us. Very early on, we learn the words “certain” and “impossible”. This is our first introduction to probability. Later on, we learn to settle disputes among our siblings and friends using games of chance. Flip a coin. Pick a number, closest wins. Which hand is the good toy in?
As we get older and learn a bit more mathematics, we can be a lot more specific in how we describe probability.
And the best way to describe this is with an equation.
Chance of something happening = number of ways it could happen / Total number of outcomes
You write probability as a percentage. “There is a 50 percent chance the coin will land on heads”. Probability can also be described as a decimal between 0 and 1. That is what you get when you use the fraction above. Most mathematical formulas assume the probability is written in this way.
The way sports odds are written is just another way to write probability. Sports odds, however, use Successes to failures (odds on) or Failures to Successes (odds against). For example, the odds that a 7 will be rolled in a pair of dice are 5 to 1 against.
Assuming that it is impossible to two events to happen at the same time, you can find the probability of either of two events happening by adding together the probabilities. For example, let’s say the probability of a particular team scoring two goals in a half is 0.1 and the probability that they will score one goal in a half is 0.3. Then the probability that they will score 1 or 2 goals in a half is 0.1 + 0.2 = 0.3.
If you want to find the probability that two events will both occur, then you multiply their probabilities. Assuming the events are independent of each other. If the events aren’t independent it gets a lot more complicated.
As you read more about probability, you will undoubtedly encounter something called Conditional probability. Conditional Probability is the probability of an event given that other events have occurred.
Consider the following example:
You have two coins. One is a normal, fair coin and the other is a two headed coin. You pick one coin. What is the probability that you picked the two headed coin?
Obvious, right? There is a 50-50 chance either way. Now let’s add some more information.
You have two coins. One is a normal, fair coin and the other is a two headed coin. You pick one coin. You flip it twice. It lands heads up both times. What is the probability that you picked the two-headed coin? Is it still 50-50? Turns out the probability is 80 percent that you chose the two headed coin.
Perhaps you are not convinced? Now what if you flipped the coin a hundred times and it landed heads up all one hundred times? Seems pretty likely that you picked the two headed coin, doesn’t it? In fact, there is a 1024/1025 chance you picked the two headed coin, which happens to be about 99.9%.
How do you do it? Count the possible outcomes. Let’s do it for a single flip of the coin. Let’s assume we picked either the two headed coin or the fair coin, flipped it once, and it landed heads-up. List the possible outcomes like this:
|Pick 2 Headed Coin||Pick Fair Coin|
We know it landed heads up, so we’ll highlight those events. Those represent the total possible outcomes, because we know it happened.
|Pick 2 Headed Coin||Pick Fair Coin|
Now we want to know the probability the coin was two-headed. It is 2 out of 3, or 0.67.
Sometimes the results of probability questions defy expectations. And they can stump even professional mathematicians. One such problem is the Monty Hall Problem, named after the host of the old American television show Let’s Make a Deal.
At the end of the show, the host offers the contestant three doors. Two contain junk, and one a fabulous prize. The contestant is asked to pick a door. Monty Hall then opens one of the other doors, showing one of the junk prizes. Now the contestant is given the choice of keeping the door they originally chose or switching to the one remaining door. Should you switch or stay? Or does it matter either way?
Marilyn Vos Savant presented this problem in one of her columns and claimed that no matter what, you will have a higher probability of winning if you switch.
Think on this for a second. At first glance, it seems like it shouldn’t matter if you switch or not. The probability should be the same. If you think that, you are in good company. Her mailbox was filled from letters from professional mathematicians claiming she was wrong.
Turns out she is very much right. The answer is found simply by using the equation at the very top of this page. Count the favorable outcomes and divide by the total number of outcomes.
When originally picking a door, the odds of picking the door with the prize is 33% (1 in 3). However after one door is revealed, the odds of the prize being behind the remaining door is 66%. This is because when the original door is chosen the two remaining door are paired together for a combined probability of 66%.
Still not quite convinced? Try this Monty Hall simulator for yourself and see the results.
This is quite a bit to take in all at once. Probability is a large subject. It is its own branch of mathematics.