### Video Transcript

In the sample space 𝑆 for events 𝐴 and 𝐵, what is the set of outcomes for the event the complement of 𝐴 intersection 𝐵?

We begin by recalling the set notation the complement of 𝐴, which can also be written 𝐴 prime. The intersection of two events is the set of outcomes in both events. From the Venn diagram, we see that set 𝐴 contains the outcomes three, six, nine, 12, and 15. These are the numbers in the three times table between one and 15 inclusive.

The complement of event 𝐴 will contain all other outcomes in the sample space that are shown on the Venn diagram. These are one, two, four, five, seven, eight, 10, 11, 13, and 14. These are the numbers not in the three times table. Event 𝐵 has three outcomes. These are the numbers four, eight, and 12, the numbers in the four times table between one and 15.

We are trying to find the intersection of these last two events. There are two numbers that appear in the complement of event 𝐴 and in event 𝐵. And these are four and eight. We can therefore conclude that the complement of event 𝐴 intersection 𝐵 contains the two elements four and eight. On the Venn diagram, this is represented by the set of elements just in set 𝐵.

This leads us to a general rule of probability, which states that the intersection of the complement of 𝐴 and 𝐵 is equal to 𝐵 minus 𝐴. Our answer will be all the elements in event 𝐵 minus those that are in event 𝐴, in this question the number 12.