Probability

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Probability is the chance that an event or outcome will occur, and it’s calculated by dividing the number of times an event happened by the number of times the event could have happened.
For example, let’s say you have one six-sided die and you want to know the probability of rolling a certain number, like a three.
Typically, probability is written with a capital P, and P of A represents the probability of “event A” happening. In this situation, event A is rolling a 6.
Since a die has six sides, there are six possible numbers you could roll, so the probability of rolling a three is 1 divided by 6, or 0.167.
Probability can be written as a decimal or as a percent, so the chance of rolling a three is 0.167 times 100 or 16.7%. Now, there are eight basic rules in probability.
The first rule states that the probability of event A can range anywhere from 0 - or 0% - to 1 - or 100%. The larger the probability is, the higher the chance that the event will occur.
The second rule states that the sum of the probabilities of all possible outcomes has to equal 1. For example, the probability of rolling each side of the die is 0.167, and when we add up 0.167 six times, it equals 1.
Sometimes we might want to find the probability that an event won’t occur - like if we wanted to figure out the probability of not rolling a three.
The probability of an event not occurring is called the complement, and it’s written as the probability of the event, except it has a prime symbol - which is just an apostrophe.
The third rule of probability states that the probability that an event doesn’t occur is 1 minus the probability that it does occur.
So, the probability of not rolling a three is 1 minus 0.167, or 0.833. Turning that around, it also means that the probability of the event occurring equals 1 minus the complement.
This is helpful in situations where we want to figure out the probability that an event occurs, but only know the probability that the event won’t occur.
Rule 4 has to do with finding the probability of two or more events happening, which is called a compound event. For example, let’s say we want to know the probability of rolling a 3 or rolling a 5.
In this case, rolling a three is the first event, or event A, and rolling a 5 is the second event, or event B, so the probability of A or B is a compound event.
Now, there are two types of compound events, and the first type is the union of two events, which is the probability of A or B occurring.
Typically it’s written with the union symbol, which looks like a small U. In a union of two events, the two events can either be disjoint - or mutually exclusive - or not disjoint - or not mutually exclusive.
So, if event A is rolling a 3 and event B is rolling a 5, then A and B are disjoint events, because it’s impossible to roll a 3 and a 5 number at the same time.
Disjoint events are often represented by two circles - one for event A and one for event B - sitting side by side, with no overlapping area.
Rule 4 is also called the addition rule and it states that the probability of two disjoint events is the sum of the first event plus the second event.
So, the probability of rolling a 3 is 0.167 and the probability of rolling a 5 is 0.167, so the probability of either rolling a 3 or a 5 is 0.167 plus 0.167, or 0.334, or 33.4%.
If two events are not disjoint, then the two events can occur at the same time. For example, if event A is rolling a number less than or equal to 2 - so rolling a 1 or 2 - and event B is rolling an even number, then A and B are not disjoint events, because it’s possible to roll a 2, which is an even number that’s less than or equal to 2.
Not disjoint events are often represented by two overlapping circles, and the overlapping area is the probability of both events occurring at the same time.
In this example, the overlapping area is the probability of rolling a 2; the non-overlapping area in circle A is the probability of rolling a 1 - which is a number that is less than or equal to 2, but not an even number; and the non-overlapping area in circle B is the probability of rolling a 4 or 6, which are even numbers but not numbers that are less than or equal to 2.
It’s a bit more complicated to calculate the probability for not disjoint events because you have to take into account the overlapping area.
So let’s break it down. The probability of rolling a number less than or equal to 2 is 2 over 6, or 0.33, and the probability of rolling an even number - so a 2, 4, or 6 - is 3 over 6, or 0.5.
If we use the basic addition rule, the probability of events A or B is 0.33 plus 0.5, which is 0.83 - or 83%. But this number is actually higher than the true probability for events A or B, because the overlapping area is counted twice.
To fix this, we need to subtract the probability of the overlapping area from the sum of the two individual probabilities.
So Rule 5 states that the probability for two not disjoint events equals the sum of the probability of event A and the probability of event B, minus the probability of event A and B together.
For example, let’s say the probability of rolling an even number that’s less than or equal to 2 is 0.167. If the probability of event A is 0.33 and the probability of event B is 0.5, then the probability of event A or event B occurring is 0.33 plus 0.5 minus 0.167, which equals 0.663, or 66.3%.
One important thing to notice is that the addition rule for disjoint events is actually the same as the addition rule for not disjoint events - except with disjoint events there is no overlapping area so you just subtract 0, and this part isn’t usually included in the equation.
Now let’s switch gears and talk about the situation when you want to figure out the probability that both event A and event B will occur - or in other words, the probability of the overlapping area.
This is the intersection of two events, and it’s the second type of compound event. Typically, the intersection of two events is written with an intersection symbol, which looks like an upside down U.
To calculate the probability of two intersecting events, you have to know if the events are independent or dependent. Two events are independent if the occurrence of the first event doesn’t affect the probability that the other event will occur.
For example, let’s say I have two dice and I want to know the probability of rolling a 3 on the first die - which is the first event - and a 5 on the second die - which is the second event.
In this case, the result of the first die doesn’t affect the result of the second die at all. Said differently, regardless of what you roll for the first die, the probability of rolling a 5 on the second die is always 1 over 6, or 0.167.
Now, to calculate the probability of rolling a 3 on the first die and a 5 on the second die, you use Rule 6 - the multiplication rule for independent events - which states that the probability of two independent events equals the probability of the first event times the probability of the second event.
In this case, both probabilities are 0.167, so 0.167 times 0.167 is 0.0278, or 2.78%. On the flip side, if two events are not independent, then the occurrence of one event affects the probability of the other event, and this is called conditional probability.
For example, let’s say you want to figure out the probability of rolling a combination of two dice that add up to 7. So, you roll two dice but this time you keep one of the die hidden under your hand and you show the other die to your friend, who tells you that it’s either a 5 or a 6.
In this situation, what we really want to figure out is the probability of event A - which is getting a combination of dice that add up to 7 - given that event B happened - which is rolling either a 5 or a 6.
Since the probability of event A depends on what happens in event B, these events are not independent, and we use conditional probability.
The way to figure out conditional probability is by using Rule 7, which states that the conditional probability - or probability of event A, given what happened in event B - equals the probability of event A and event B divided by the probability of event B - and in this equation, the word “given” is typically replaced by a vertical bar symbol.
The first step in finding the conditional probability is figuring out the probability of both events A and B. To find the probability of event A - getting a combination of dice that add up to 7 - we have to figure out how many different combinations of numbers there can be with two dice.
This can be shown in an outcome table, where the possible outcomes for the first die are on top and the possible outcomes for the second die are on the side, and the middle numbers show the sum of both dice.
So, there’s a number 7 in the upper right square because if you roll a 1 on the first die and a 6 on the second die, the sum equals 7.
The total number of squares in the outcome table is 36, and that’s also the total number of possible outcomes from the sum of 2 dice.
A simple way to figure this out is to multiply the number of outcomes possible for the first die - so 6 - by the number of outcomes possible for the second die - which is also 6 - which equals 36.
Now that you know the total number of combinations for two dice, you have to figure out the number of combinations that equal 7.
Looking at the table, you can see that there are six combinations for the number 7. You could get a 1 on the first die and a 6 on the second die; a 2 on the first die and a 5 on the second die; or a 3 on the first die on a 4 on the second die.
On the other hand, you could get the opposite for each combination - so a 6 and a 1, a 5 and a 2, or a 4 and a 3. Since there are 6 possible combinations that add up to 7 and 36 total combinations possible, the probability of event A is 6 over 36, or 0.167.
For the probability of event B, there are 2 possible outcomes - if you rolled a 5 or if you rolled a 6 - and there are 6 total possible outcomes, because you could have rolled any of the numbers on the die.
So, the probability of event B is 2 over 6, or 0.33. Now that you have both the probabilities for event A and event B, you can plug them into the conditional probability equation.
To find the probability of events A and B, you use the multiplication rule, so 0.167 times 0.33 is 0.055. Then, divide 0.055 by the probability of event B - or 0.33 - to get 0.167.
You can also figure out the conditional probability of event B, given that event A occurred. In other words, you could calculate the probability that one of the die was a 5 or a 6 given that the sum of the two dice equals 7.
To do this, you would still multiply the probability of A - or 0.167 - times the probability of B - or 0.33 - but this time you would divide by the probability of A.
So 0.167 times 0.33 is 0.055, divided by 0.167, equals 0.33. Now let’s take it one step further.
We can use conditional probability to calculate the probability of event A and event B when the two events are not independent, and this is Rule 8 - the multiplication rule for dependent events.
This is different from Rule 7, conditional probability, which is the probability of event A given event B occurred. In fact, the multiplication rule for dependent events uses the conditional probability, and it states that the probability of events A and B equals the probability of event A times the conditional probability of event B given event A occurred.
So, to figure out the probability of events A and B - which would be the probability of getting a sum of two dice that equals 7 and rolling either a 5 or a 6 - you would multiply the probability of event A - getting a sum of two dice that equals 7 - which was 0.33, and the conditional probability - rolling a 5 or a 6 given that the sum of the two dice was 7 - which was also 0.167.
So, the probability of events A and B is 0.33 times 0.167, or 0.055. In these examples, we had all the information about each one of the probabilities, but in real life this oftentimes isn’t the case.
Alright, as a quick recap. There are eight basic rules of probability.
The first rule says that probability of event A can range anywhere from 0 to 1. The second rule says that the sum of the probabilities of all possible outcomes has to equal 1.
The third rule says that the probability that an event doesn’t occur is 1 minus the probability that it does occur. The fourth rule says that the probability of two disjoint events is the sum of the probability of event A and the probability of event B.
The fifth rule says that the probability of two not disjoint events is the sum of the probability of event A and the probability of event B minus the probability of A and B together.
The sixth rule says that probability of two independent events equals the probability of the first event times the probability of the second event.
The seventh rule says that probability of event A, given what happened in event B equals the probability of event A and event B divided by the probability of event B.
Finally, the eighth rule says that the probability of two dependent events equals the probability of event A times the conditional probability of event B given event A.