Many students don’t learn basic probability rules until they take a statistics class, or maybe as late as college. However, these rules are important for the Math sections of the SAT and ACT, and they can also be useful in our everyday lives.
Probabilities As Decimals, Fractions, and Percentages
Most people are used to thinking of probabilities as percentages, but a probability can be expressed in multiple ways. For example, 18% is the same as 0.18, which is the same as 9/50 and 18/100 . These are all equally correct.
Nonetheless, it is often more useful mathematically to treat probabilities as decimals or fractions. Students in statistics classes or taking the SAT or ACT will probably want to adopt this habit as well.
Mutually Exclusive Events
The first probability rule to be aware of is determining whether two events are “mutually exclusive.” This means that the two events cannot possibly both occur. For example, imagine that you flip a coin one time, and you define event H as getting Heads and event T as getting Tails. Events H and T are mutually exclusive, because a coin can’t land on both sides at once.
Conversely, imagine a scenario where event A represents wearing a black shirt on a given day, and event B represents wearing black pants on a given day. Events A and B are not mutually exclusive, because doing one doesn’t stop you from doing the other.
Independent Events
Two events are independent if the occurrence of one event does not affect the occurrence of the other.
For example, imagine a hypothetical town where 60% of families own a house, and 35% of families have a pet dog. Of those families that own a house, 55% have a pet dog. This is an example of events that are not independent, because knowing that a family owns a house changes the probability that the family has a dog.
Mathematically, we can test for independent events by using the formula below. Note that only one of these needs to be tested—we do not need to test both of them.
if P(A | B) = P(A), then A and B are independent
if P(B | A) = P(B), then A and B are independent
In this context, the “|” symbol indicates “given that.” If we call event A “owning a house,” event B “owning a dog,” and event B | A “owning a dog given that you own a house,” then we have the following values:
P(A) = 0.60
P(B) = 0.35
P(B | A) = 0.55
Because the second and third values are not equal, we can conclude that owning a house and having a dog are not independent events.
P(B) ≠ P(B | A)
0.35 ≠ 0.55
Additive Rule (Mutually Exclusive Events)
Perhaps the most intuitive probability rule for many students, the additive rule states that, when two events are mutually exclusive, the probability of one event OR the other event occurring is equal to the sum of their probabilities. Formally, this is shown in the formula below.
P(A or B) = P(A∪B) = P(A) + P(B), when A and B are mutually exclusive
For example, let’s imagine that we have a deck of cards, we are going to draw a single card from the deck, and we want to select a Queen OR a red King. Because these are mutually exclusive events (it is not possible to select a Queen and a red King on the same draw), we can find the probability of selecting a Queen OR a King by adding the probabilities together:
4/52 + 2/52 = 6/52 = 3/26 ≈ 0.1154
This relationship also holds true when there are more than two events. Now imagine that you wanted to know the probability that you select a Queen OR a red King OR the Ace of Spades. Because these are still mutually exclusive, we can still add the probabilities.
4/52 + 2/52 + 1/52 = 7/52 ≈ 0.1346
Additive Rule (Events that are not Mutually Exclusive)
When two events are not mutually exclusive, we unfortunately can’t simply add all of the separate probabilities. For example, imagine that a hypothetical high school is 51% female, and that 87% of its students are at least 15 years old. If we simply added these probabilities, we would get 0.51 + 0.87 = 1.38. In other words, 138% of the students are female or at least 15 years old, which just doesn’t make sense, since no probability can be above 100%.
So, why doesn’t this work? Well, it’s because the two groups overlap: some females at the school are at least 15, and vice versa. Because of this, we’re counting some people twice. The way we fix this is subtract the overlap.
P(A or B) = P(A) + P(B) – P(A and B)
In our example, let’s assume that 44% of students at the school are females who are also at least 15 years old.
P(Female or at least 15 years old) = 0.51 + 0.87 – 0.44 = 0.94
Now the result makes much more sense: the probability of a randomly selected student being female OR at least 15 years old is 0.94, or 94%.
Multiplicative Rule
What if we want to know the probability that events A and B both occur, instead of the probability that one or the other occurs? This is where the multiplicative rule comes in. This rule says that, for two independent events, the probability of both occurring is equal to the product of their separate probabilities.
P(A and B) = P(A) * P(B), when A and B are independent
For example, imagine that you have 7 hats (4 of which are red) and 9 pairs of gloves (8 of which are black). If you randomly select one hat and one pair of gloves, what is the probability that you select a red hat AND a black pair of gloves?
P(red hat AND black gloves) = 4/7 * 8/9 = 32/63 ≈ 0.5079
Conclusion
Now we know how to use some of the basic probability rules! In a future post, we’ll cover some more advanced probability rules, including conditional probability, how to handle dependent events, and more. See you then!
