### Video Transcript

For two events 𝐴 and 𝐵, the probability of 𝐵 is two-fifths, the probability of 𝐴 intersection 𝐵 is one-tenth, and the probability of 𝐴 union 𝐵 is three-quarters. Work out the probability of 𝐴.

In order to answer this question, we will 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 told in the question that the probability of 𝐵 is two-fifths, the probability of 𝐴 intersection 𝐵 is one-tenth, and the probability of 𝐴 union 𝐵 is three-quarters. Substituting these into the addition rule of probability, we have three-quarters is equal to the probability of 𝐴 plus two-fifths minus one-tenth.

The denominators of the three fractions four, five, and 10 are all factors of 20. Multiplying the numerator and denominator of three-quarters by five, we get 15 over 20. We can use a similar method for the other two fractions such that all the three have a common denominator of 20.

Eight over 20 minus two over 20 is six over 20. So our equation simplifies to 15 over 20 is equal to the probability of 𝐴 plus six over 20. We can then subtract six over 20 or six twentieths from both sides such that the probability of 𝐴 is equal to 15 over 20 minus six over 20. This is equal to nine over 20 or nine twentieths.

For the two events 𝐴 and 𝐵, where the probability of 𝐵 is two-fifths, the probability of 𝐴 intersection 𝐵 is one-tenth, and the probability of 𝐴 union 𝐵 is three-quarters, then the probability of 𝐴 is nine twentieths.