Question Video: Using the Addition Rule to Determine the Probability of Union of Two Events Mathematics • 10th Grade

Video Transcript

Suppose that 𝑋 and 𝑌 are two events with probabilities 𝑃 of 𝑌 equals one-third and 𝑃 of 𝑋 equals 𝑃 of the complement of 𝑋. Given that 𝑃 of 𝑋 intersects 𝑌 equals one-eighth, determine 𝑃 of 𝑋 union 𝑌.

In this question, we’re given two events 𝑋 and 𝑌. And we are told that the probability of event 𝑌 occurring is one-third and that the probability of event 𝑋 occurring is equal to the probability of the complement of 𝑋 occurring. We’re also told that the probability of the intersection of these two events is one-eighth, that is, the probability of both events occurring. We want to use this to determine the probability of the union of these events, that is, the probability that either event occurs.

To find this probability, we can start by recalling that the addition rule for probability tells us that for any events 𝑋 and y, 𝑃 of 𝑋 union 𝑌 is equal to 𝑃 of 𝑋 plus 𝑃 of 𝑌 minus 𝑃 of 𝑋 intersects 𝑌. We’re given two of these probabilities in the question. So we can determine the probability of 𝑋 union 𝑌 if we can determine the probability of 𝑋. We can find the probability of 𝑋 by using the fact that 𝑃 of 𝑋 is equal to 𝑃 of the complement of 𝑋. That is, the chance of 𝑋 occurring is equal to the chance it does not occur.

This is enough to conclude that the probability of 𝑋 occurring is one-half. However, it can be useful to show this result formally. We can recall that for any event 𝑋, the probability of 𝑋 occurring is equal to one minus the probability it does not occur. We are told that 𝑃 of the complement of 𝑋 is equal to 𝑃 of 𝑋, so we can substitute this into the equation. This gives us that 𝑃 of 𝑋 equals one minus 𝑃 of 𝑋. We can now solve for 𝑃 of 𝑋.

We add 𝑃 of 𝑋 to both sides of the equation to get two 𝑃 of 𝑋 equals one. Then, we divide both sides of the equation by two to obtain 𝑃 of 𝑋 equals one-half. We can now substitute this value into the addition formula for probability to determine the probability of either event 𝑋 occurring or event 𝑌 occurring. This gives us that 𝑃 of 𝑋 union 𝑌 is equal to one-half plus one-third minus one-eighth. We can now evaluate this expression by writing each probability with a common denominator of 24. We have 12 over 24 plus eight over 24 minus three over 24. We can then evaluate this expression to obtain 17 over 24, which is our final answer.