Probability of Dependent and Independent Events
Episode #2 of the course Theory of probability by Polina Durneva
Today, we will talk about the probability of dependent and independent events and conditional probability. These concepts are closely related and based on each other.
Probability of Independent Events
First, let’s define independent events. Independent events are those that do not affect each other. In the previous lesson, we talked about mutually exclusive events. Sometimes, people confuse these two ideas, so it would be useful to clarify these two terms one more time. Mutually exclusive events are the events that cannot occur at the same time. For instance, if you have to make a choice between a burger and a hot dog and cannot buy two of them at the same time, these are mutually exclusive events. Independent events can happen simultaneously or in a sequence. For instance, your decision of what to wear tonight is not affected by what coffee your cousin drank in the morning. These two events are completely unrelated and thus, independent.
If you want to calculate the probability of independent events happening all together, you just need to multiply the probabilities of each individual independent event.
The formula takes the following form: Probability(Event A & Event B) = Probability(Event A) * Probability(Event B).
For example, let’s assume that you are about to toss a coin 3 times, and you want to know the probability of getting 3 heads in a row. What is the probability of getting a head? Well, since you have only two options, either heads or tails, the probability is 1/2, or 50%. Therefore, the probability of getting 3 heads in a row is: (1/2) * (1/2) * (1/2) = 1/8 = 0.125 = 12.5%.
Probability of Dependent Events
Dependent events, as you may guess, are events whose occurrence affect each other. As for probabilities, it means that the probability of one event impacts the probability of the other event. For example, your decision to take an umbrella is probably affected by the weather outside.
The formula for the probability of dependent events is slightly more complicated than that for the probability of independent events. It takes the following form for two dependents, events A and B: Probability(Event A & Event B) = Probability(Event A) * Probability(Event B | Event A). All the components should look familiar except for the last term. Probability(Event B | Event A) is denoted as the probability of Event B given that Event A has already occurred. This term is called conditional probability.
To better understand the concept, let’s proceed to an example. Let’s assume you have a class of students. Fifty of your students are biology majors and 30 of your students are computer science majors. First, you randomly select one student out of 80. Then, you select another student out of 79. What is the probability that both of these selected students are computer science majors?
Let’s first solve this problem without a formula. You have to pick your first student. The probability that this student is a computer science major is 30/80. After you pick your first student, you have 79 students left (and supposedly, 29 computer science majors are left). The probability of the 2nd student to be a computer science major is 29/79. You can calculate the total probability by multiplying the two terms: (30/80) * (29/79) = 13.77%.
As for formula, it takes the following form: Probability(2 students are computer science majors) = Probability(1st student is a computer science major) * Probability(2nd student is a computer science major | 1st student is a computer science major). Probability(1st student is a computer science major) is equal to 30/80. The second term, the probability of the 2nd student being a computer science major, is 29/79. The multiplication of these two terms is equal to 13.77%.
That’s it for today. Tomorrow, we’ll proceed to more complicated material by discussing our first discrete probability distribution: Bernoulli distribution.
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