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