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Probability is a whole branch of mathematics that is interested in describing how likely it is that something will occur. Oftentimes, we limit our understanding of probability to dice, cards, and other games of chance. However, you can apply probability to statements!

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For instance, if I say that “pigs are purple,” you’ll most likely tell me that the probability of that statement being true is 0, since you’ve never seen (or likely heard of) a purple pig before. However, if someone were to color a pig purple with paint, then the statement could easily be true!

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This shows that probability can extend to concepts beyond simple games of chance.

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On your standardized tests and in most classes, you’ll most often be working with probability dealing with numeric quantities.

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For this kind of probability, you will be dealing with standard models of dice, cards, marbles in a bag, flipping a coin, etc…

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This kind of probability can be measured by the number of ways that something can occur divided by the total number of outcomes.

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A single die has 6 sides, labeled from 1 to 6. If I ask you “what is the probability that we roll an even number?”, then we can easily calculate the answer using this formula!

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First, count the number of even numbers on the die. The die has 2, 4, and 6, which are all even. Therefore, there are 3 even numbers.

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Second, count the total number of sides of the die. We know that there are 6 sides, which is the total number of outcomes.

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Finally, we can write out the probability to be 3/6 = 1/2.

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This also makes sense when we just think about a die in general. Exactly half of the numbers on the die are even, so the probability of choosing an even number is 1/2!

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There are two key words that greatly influence the probability of something happening:

- And
- Or

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When a probability problem statement uses the word “and”, we must multiply probabilities together.

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When a probability problem statement uses the word “or”, we will end up adding probabilities together.

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There are two other things we must also consider with each of these words!

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For “and”, we must consider whether our events are

- Independent
- Dependent

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If the events are independent, then the outcome of one event does not affect the other. This is like if you roll a die and draw a card from a deck of cards at the same time. These two events don’t impact each other.

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However, if you draw a card from a deck of cards, don’t return it and then draw another card, you’ve changed the probability of the second card! This is known as a dependent event, since it depends on the earlier event.

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For “or”, we must consider whether our events are

- Mutually exclusive
- Not mutually exclusive

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This sounds incredibly similar to independent versus dependent, but there’s a subtle difference. Two events are mutually exclusive when they don’t overlap at all. For instance, if you roll a single die, you can’t roll both a 3 and a 4 at the same time. These are mutually exclusive events.

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To contrast this, events that are not mutually exclusive have some overlap. For instance, if you roll that same die again, we can ask about the probability that you roll a number less than 5 or you roll an even number. We can see that these events overlap, since 2 and 4 are both less than 5, but they are also even numbers! This is much better understood with a Venn diagram.

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Now that we’ve gotten all of that technical jargon out of the way, let’s work on some applications!

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Suppose you are asked about the probability of drawing a Jack and then a Queen from a deck of cards. If we return the first card we draw, then these are actually independent events. There are 52 cards in a standard deck of cards and there are 4 Jacks and 4 Queens.

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Since the word “and” was used in the original problem, we must multiply the probability that we draw a Jack times the probability that we draw a Queen. This is 4/52 × 4/52 = 1/13 × 1/13 = 1/169.

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Notice that the probability of drawing a Jack and drawing a Queen is the exact same because we didn’t change the deck of cards at all after drawing the first card.

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Now, let’s suppose we didn’t return the Jack after we drew it the first time. This would create dependent events.

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The Jack has the same probability of 4/52, but there are only 51 cards left after we remove the Jack. This means that drawing a Queen after the Jack has a probability of 4/51!

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Multiply these together to find that the probability of drawing a Jack and then a Queen without replacement is 4/52 × 4/51 = 1/13 × 4/51 = 4/663 which is approximately 1/166.

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Notice how much more likely it is to draw a Jack and then a Queen when you don’t replace the cards! This is what makes probability so fascinating to study.

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Finally, we’ll consider a probability problem using “or”.

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A classroom of 20 students was polled as to what their favorite class in school was: history or math.

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6 students responded that math was their favorite class, 4 students responded that history was their favorite class, and 10 students said neither of these classes was their favorite class. What is the probability that a student in this class responded that math or history was their favorite class?

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To solve this problem, we must add the probability that a student’s favorite class is math with the probability that the student’s favorite class is history.

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Therefore, we know that the total probability is 6/20 + 4/20 = 10/20 = 1/2. Half of the students’ favorite class was math or history!

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Probability can become even more complex than presented here and it is one of the most interesting (and least taught) mathematics! Try to apply probability to your everyday life and see if you can calculate the odds of basic events occurring.

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