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.
