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Competition Math

Probability

What is Probability? 

There’s a great chance that you have heard of probability before. Speaking about chances, that’s exactly what probability is! Probability is the extent to which an event is likely to occur. Well, if that’s the case, then how do you calculate probability? Probability is measured by the ratio of the favorable cases to the whole number of cases possible.

Let’s start off with a basic example: What is the probability of getting a “heads” when flipping a regular coin? Well, as we said previously, probability is measured by the ratio of the favorable cases to the whole number of cases possible. So what’s the favorable cases in this example? Well, we are trying to get heads, so that would be the favorable case. Thus, we would have only 1 favorable case. The next part of what we have to look at is the whole number of cases possible. Well, in a regular coin, how many possibilities do we have? Well, we have heads and tails, so just 2 possibilities. Thus, we will take the ratio of favorable to possible cases and get 1:2. In other words, that means that there is a ½ probability of getting a heads when flipping a regular coin. 

Independent vs Dependent Events

Now, in probability, there are two types of events that can happen: independent and dependent events. The example we just did above is an example of an independent event. An independent event is when 2 separate events happen, and the first event does not impact the result of the second event. For example, as we saw in this example, we flipped a coin to see the probability of getting a heads. Whether we got heads or tails on the first flip, it does not affect the result of the following flip. On the other hand, a dependent event is when the outcome of the first event affects the probabilities of the subsequent events. 

Calculating Independent and Dependent Probabilities

When it comes to calculating Independent and Dependent Probabilities, it is very simple. For independent probability, the probability that both the events that will occur is the probability of event A happening times the probability of event B happening: P(A) x P(B). But make sure to identify whether the event is independent or dependent by seeing if the event 

For dependent probability, the probability that event A and B will happen is equal to the probability of event A happening times the probability of event B given A: P(A and B) = P(A) x P(B|A). Remember that for dependent probability, the result of the first event happening affects the outcome of the second event happening, so we will most likely have a different number of possible outcomes after the first event has happened. 

Why Probability?

Okay, that’s great that we know all of this, but how will this actually help us? Well, probability is used all around us. For example, in the morning, when you are getting ready for school, you dress according to the weather. If it is a sunny and hot day, you are not going to wear a jacket. You are most likely going to wear something more suitable for the weather such as shorts. Although you might not know it, you are a mathematician in your sleepy state in the morning. Another reason why you may want to learn about probability is because it is one of the hot topics throughout school and in competition math. For example, in the 2020 AMC 8 Math competition, more than 25% of the problems dealt with probability! That’s a big portion of the test right there, and if you are comfortable with probability, it will make it that much easier for you to do those problems. In addition to just solving problems, probability can provide you with so many interesting facts. For example, probability can let us know the odds of someone getting attacked by a shark: 1 in 11,500,000. Or, probability can tell us the odds of someone becoming the president of the United States: 1 in 10,000,000. 

Probability is incredibly interesting, and you start researching it by going to the resources provided below. These resources will give you much of the essential information you will need to become a probability genius!

Resources

https://www.mathsisfun.com/data/probability.html (A great tool to learn about what probability is, how to calculate it, and some examples to get you familiar with probability)

https://www.mathgoodies.com/lessons/vol6/independent_events (A very helpful website to learn more about independent events and some example problems as well)

https://www.mathgoodies.com/lessons/vol6/dependent_events (A very informative website teaching all about dependent events and examples)

https://artofproblemsolving.com/wiki/index.php/Probability (AOPS’s page on probability and examples from previous competitions)

A fascinating video that goes over the basics of probabilities, more specifically, permutations and combinations.
A great video resource that focuses on casework, complementary counting, and overcounting.

https://www.onlinemathlearning.com/probability-problems.html (A great source of example problems, with solutions, on many topics in probability)

https://artofproblemsolving.com/wiki/index.php/Category:Introductory_Probability_Problems (AOPS’s breakdown of all the introductory probability problems in all previous math competitions)