# Question Video: Using the Addition Rule to Solve Problems in Probability Mathematics

Students at a school must wear a sweatshirt or a blazer, and are allowed to wear both. In one class of 32 students, 12 students wear blazers, and 4 of the students who wear a blazer also wear a sweatshirt. Let 𝐴 be the event of randomly selecting a student from the class who wears a blazer and let 𝐵 be the event of randomly selecting a student from the class who wears a sweatshirt. Find 𝑃(𝐴). Find 𝑃(𝐵). Find 𝑃(𝐴 ⋂ 𝐵). Find 𝑃(𝐴 ⋃ 𝐵).

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### Video Transcript

Students at a school must wear a sweatshirt or a blazer and are allowed to wear both. In one class of 32 students, 12 students wear blazers, and four of the students who wear a blazer also wear a sweatshirt. Let 𝐴 be the event of randomly selecting a student from the class who wears a blazer and let 𝐵 be the event of randomly selecting a student from the class who wears a sweatshirt. There are four parts to this question. Find the probability of 𝐴, find the probability of 𝐵, find the probability of 𝐴 intersection 𝐵, and find the probability of 𝐴 union 𝐵.

In all four cases, we’re asked to give our answer as a fraction in its simplest form. We could work out the first and third parts directly from the question. We are told that 12 out of the 32 students wear blazers; therefore, the probability of 𝐴 is 12 out of 32. As both the numerator and denominator are divisible by four, this simplifies to three out of eight or three-eighths. We are also told that four of the students who wear a blazer also wear a sweatshirt. This means that the probability of 𝐴 intersection 𝐵 is four out of 32, as this is the probability of selecting a student who wears a blazer and a sweatshirt. Once again, we can divide the numerator and denominator by four, giving us one-eighth.

A key word in the question is “must,” as we’re told that students must wear a sweatshirt or a blazer. We can use this fact to write down the probability of 𝐴 union 𝐵. As all of the students must wear a sweatshirt or a blazer or both, the probability of 𝐴 union 𝐵 is 32 over 32, which is equal to one. It is certain that a randomly selected student will be wearing at least one of a sweatshirt or a blazer.

This leaves us with the second part of the question, calculating the probability of 𝐵, the event of randomly selecting a student who wears a sweatshirt. We can answer this by using the addition rule of probability, which states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. Substituting in the values we already know, we have one is equal to three-eighth plus the probability of 𝐵 minus one-eighth. The right-hand side simplifies to two-eighths plus the probability of 𝐵, which in turn is equal to a quarter plus the probability of 𝐵. Subtracting one-quarter from both sides of this equation gives us the probability of 𝐵 is equal to three-quarters.

We now have answers to all four parts of this question. They are three-eighths, three-quarters, one-eighth, and one, respectively. We could also represent the information on a Venn diagram, where the numbers shown are the number of students in each section. There were 12 students that wear blazers, 24 students that wear sweatshirts, and four students that wear both. The three numbers sum to give us a total of 32 students.