Question Video: Using Venn Diagrams to Calculate Probability

Video Transcript

Isabella has drawn this Venn diagram to record the result of randomly selecting a number between one and 12. What is the probability of selecting a number that is a factor of 20? What is the probability of selecting a number that is a factor of 20 and a multiple of three? What is the probability of selecting a number that is not a multiple of three?

We note that our Venn diagram is made up of two circles, firstly the numbers that are factors of 20. We will call this event 𝐴. Secondly, we have the circle that contains all the multiples of three. We will call this event 𝐵. The numbers seven, eight, and 11 that are outside of both circles are the numbers between one and 12 that are not factors of 20 and also not multiples of three.

The first part of our question wants us to calculate the probability of selecting a number that is a factor of 20. This is the probability of event 𝐴 occurring. We note that the numbers one, two, four, five, and 10 are all factors of 20. This means that five of the 12 numbers between one and 12 are factors of 20. The probability of selecting a number that is a factor of 20 is, therefore, equal to five-twelfths or five out of 12. As five and 12 have no common factor apart from one, this fraction is in its simplest form.

The second part of our question asks us to calculate the probability of selecting a number that is a factor of 20 and a multiple of three. This is the probability of event 𝐴 and event 𝐵 occurring, written as probability of 𝐴 intersection 𝐵. There are no numbers in the intersection of the two circles. This means that the probability of 𝐴 intersection 𝐵 or 𝐴 and 𝐵 is zero out of 12 or zero. The probability of selecting a number that is a factor of 20 and a multiple of three is zero. This is because there are no numbers between one and 12 that satisfy both of the criteria.

This leads us to an important fact when dealing with probability. If the intersection of two events is equal to zero, then the two events themselves are mutually exclusive. In this case, we could actually draw the two circles separately as there doesn’t need to be any intersection. There is no value that is a factor of 20 and a multiple of three.

The final part of our question wants us to calculate the probability of selecting a number that is not a multiple of three. This can be written as the probability of 𝐵 bar or the probability of 𝐵 prime. The probability of any event not occurring is equal to one minus the probability of that event occurring. We know that there are four numbers that are multiples of three: three, six, nine, and 12. This means that eight of the numbers between one and 12 will not be multiples of three. These are the numbers one, two, four, five, and 10 — that are factors of 20 — and the numbers seven, eight, and 11 — that are neither factors of 20 nor multiples of three.

The probability of selecting a number that is not a multiple of three is eight out of 12 or eight-twelfths. Both of these numbers have a factor of four, so we can divide the numerator and denominator by four. The fraction eight-twelfths simplifies to two-thirds. This means that the probability of selecting a number that is not a multiple of three in its simplest form is two-thirds.