In the sample space 𝑆, for events 𝐴 and 𝐵, what is the set of outcomes for the event 𝐴 bar?
The event 𝐴 bar means those values that are not in 𝐴. In this question, we have five values — three, six, nine, 12, and 15 — that are in event 𝐴. This means that all the other values — four and eight that are in event 𝐵, along with one, two, five, seven, 10, 11, 13, and 14 that are neither in event 𝐴 or event 𝐵 — are not in 𝐴. The correct set of outcomes is therefore the 10 values one, two, four, five, seven, eight, 10, 11, 13, and 14.
Whilst we’re not asked to in this question, we could use the numbers to work out what the events 𝐴 and 𝐵 are. The sample space S contains the integers from one to 15. Event 𝐴, the numbers three, six, nine, 12, and 15 are the multiples of three. Event 𝐵, on the other hand, four, eight, and 12 are the multiples of four between one and 15. The number 12 occurs in both events as it is a multiple of three and a multiple of four. The eight numbers outside of the circles are neither multiples of three or multiples of four. This means that the set of outcomes 𝐴 bar are the set of numbers that are not multiples of three between one and 15.