الأسبوع 11 · العمليات على الأحداث والأحداث المتنافية واحتمال الحدث عندما تتساوى فرص النواتج · 3 دروس
2
Mutually exclusive events: A ∩ B is empty, with two dice and with success and failure
Statistics Sec 3 Term 1, Unit 3, Lesson 1
GoalDecide whether two or more events are mutually exclusive by looking for a common outcome.
New words
mutually exclusive events · events that cannot occur together in one trial: A ∩ B = ∅simple event · an event with one outcome onlyexclusive in pairs · every two of the events are exclusive
Explanation
Two events are mutually exclusive when the occurrence of one makes the other impossible in the same trial, that is, when they have no outcome in common: A ∩ B = ∅. On one throw of a die, “a prime number” and “six” are exclusive, because 6 is not prime; “even” and “six” are not, because 6 belongs to both. Several events are exclusive when every two of them are. Two facts are worth keeping: simple events are always exclusive (a trial gives one outcome), and an event is always exclusive with its complement. The converse is false: exclusive events need not be complements. The grid shows two dice. Let A = “the sum is 9”, B = “both dice show the same number” and C = “the sum is 10”. The sum 9 appears in four cells, and the doubles run down the diagonal, whose sums 2, 4, …, 12 are all even, so A ∩ B = ∅. A and C are exclusive too, since one sum cannot be both 9 and 10. But B and C share the pair (5, 5), so B and C are not exclusive.
Sum of the two dice (first die down, second across)
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Worked examples
One exam, one pupil: “passes” and “fails” are exclusive (nobody both passes and fails the same exam), and they are also complements. Compare a die: A = {1, 2, 3} and B = {4, 5, 6} are exclusive and complements, but A = {1, 2} and B = {5, 6} are exclusive and not complements, because 3 and 4 belong to neither.
On a twelve-sector spinner let A = {1, 2, 3}, B = {4, 5, 6} and C = {6, 7, 8}. A and B share nothing, and so do A and C, but B and C share 6. So A, B and C are not exclusive as a group, although two of their three pairs are.
Pair
Common outcomes
Exclusive?
A, B
—
yes
A, C
—
yes
B, C
5
no
A, D
2
no
Pairs on one throw of a die. Let A = “less than 3” = {1, 2}, B = “a multiple of 5” = {5}, C = “greater than 4” = {5, 6} and D = “even” = {2, 4, 6}. For each pair look for a common outcome.
Class activity
Draw the 6 × 6 grid of sums of two dice. Colour the cells with sum 5 in one colour and the cells with sum 7 in another. Are “the sum is 5” and “the sum is 7” exclusive? Then colour the cells in which the first die shows 1: is “the first die shows 1” exclusive with “the sum is 5”?
Practice
1
On one throw of a die let A = “a prime” = {2, 3, 5}, B = {6} and C = “even” = {2, 4, 6}. Which pairs are exclusive?
Only A and B. A and C share 2, and B and C share 6.
2
Two coins are tossed. Are “two heads” and “two tails” exclusive? Are “at least one head” and “at least one tail” exclusive?
The first pair is exclusive (HH and TT have nothing in common). The second pair is not: HT and TH belong to both events.
3
Two dice: are “the sum is 4” and “the first die shows 1” exclusive?
No. The pair (1, 3) has sum 4 and a 1 on the first die, so it belongs to both events.
4
Karim says, “Exclusive events must be complements.” Give an example that shows he is wrong.
On a die, A = {1} and B = {2} are exclusive, but A ∪ B = {1, 2} is not S, so they are not complements.
5
Cards numbered 1 to 10: A = {5, 10}, B = {1, 4, 9}, C = {2, 3, 5, 7}. Which pairs are exclusive? Are A, B, C exclusive as a group?
A and B are exclusive, and so are B and C; A and C share 5. So the three are not exclusive as a group.
