### Video Transcript

Suppose 𝐴 and 𝐵 are two events
with probabilities the probability of 𝐴 is five-sevenths and the probability of 𝐵
is four-sevenths. Given that the probability of 𝐴
union 𝐵 is six-sevenths, determine the probability of 𝐴 minus 𝐵.

In this question, we are asked to
calculate the probability of the difference of two events. We recall that the probability of
𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of 𝐴
intersection 𝐵. This is known as the difference
rule of probability. And in this question, we are given
the probability of 𝐴 is five-sevenths. At present, we are not given the
probability of 𝐴 intersection 𝐵. Before trying to calculate this, it
is worth noting that we can represent this situation on a Venn diagram. We are trying to calculate the
probability of being in event 𝐴 but not event 𝐵.

In order to calculate the
probability of 𝐴 intersection 𝐵, we recall the addition rule of probability. This 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. We are given three of these values
in the question. The probability of 𝐴 union 𝐵 is
six-sevenths, the probability of 𝐴 is five-sevenths, and the probability of 𝐵 is
four-sevenths. By rearranging and then simplifying
this equation, we can calculate the probability of 𝐴 intersection 𝐵. This is equal to five-sevenths plus
four-sevenths minus six-sevenths. As the denominators of the three
fractions are the same, we simply add and subtract the numerators. Five plus four minus six is equal
to three. The probability of 𝐴 intersection
𝐵 is three-sevenths.

We can now substitute this value
back into the formula to calculate the probability of 𝐴 minus 𝐵. This is equal to five-sevenths
minus three-sevenths, which gives us a final answer of two-sevenths.