Video Transcript
In each case, decide whether the two events are mutually exclusive or not. Event 𝐴: Rolling a six-sided die and getting a number greater than four. Event 𝐵: Rolling a six-sided die and getting an odd number.
There are two other parts to this question that we will look at later. Let’s begin by recalling our definition of mutually exclusive events. Two events are said to be mutually exclusive if they cannot happen at the same time. This means that the intersection of two mutually exclusive events 𝐴 and 𝐵 is the empty set. There are no elements that occur in event 𝐴 and event 𝐵. When dealing with probability, the probability of 𝐴 intersection 𝐵 for two mutually exclusive events is zero.
In this part of the question, event 𝐴 involves getting a number greater than four when rolling a six-sided die. This is the set containing the numbers five and six, as shown on the Venn diagram. Event 𝐵 involves getting an odd number when rolling a six-sided die. This is the set of numbers one, three, and five, as shown on the Venn diagram by the pink circle. We notice that the number five appears in both sets. This means that 𝐴 intersection 𝐵 is the set containing the number five and is not equal to the empty set. We can therefore conclude that events 𝐴 and 𝐵 are not mutually exclusive.
We will now clear some space and consider events 𝐶 and 𝐷. Event 𝐶 involves rolling an eight-sided die and getting a number less than four. And event 𝐷 involves rolling an eight-sided die and getting a number greater than four. Once again, we are given a Venn diagram representing these two events and need to decide whether they are mutually exclusive or not. Event 𝐶 is the set of numbers one, two, and three. Event 𝐷 is the set of numbers five, six, seven, and eight. We notice that the number four is in neither event 𝐶 nor event 𝐷, as it is neither less than nor greater than four. The two circles in the Venn diagram do not intersect, as there is no number that appears in event 𝐶 and event 𝐷. 𝐶 intersection 𝐷 is therefore equal to the empty set. And we can conclude that the two events are mutually exclusive.
Let’s now consider the final part of this question. Event 𝐸 involves rolling a 20-sided die and getting a prime number greater than three. And event 𝐹 involves rolling a 20-sided die and getting a factor of 15. In this part of the question, we have not been given a Venn diagram, but we still need to work out whether the two events are mutually exclusive or not.
We recall that a prime number is any number that has exactly two factors. The prime numbers that are less than 20 are two, three, five, seven, 11, 13, 17, and 19. Event 𝐸 involves getting a prime number greater than three. This means it is the set of six numbers five, seven, 11, 13, 17, and 19. Event 𝐹 involves rolling a 20-sided dice and getting a factor of 15. 15 has two factor pairs: one and 15 and three and five. This means that set 𝐹 contains the four numbers one, three, five, and 15.
To determine whether the events 𝐸 and 𝐹 are mutually exclusive, we’ll consider the intersection of these events. The number five appears in both set 𝐸 and set 𝐹. This means that 𝐸 intersection 𝐹 is the set containing the number five. As this is not equal to the empty set, we can conclude that events 𝐸 and 𝐹 are not mutually exclusive.
In this question, we found that events 𝐴 and 𝐵 were not mutually exclusive. Events 𝐶 and 𝐷 were mutually exclusive. And events 𝐸 and 𝐹 were not mutually exclusive.
