Logical Relationships among Sets
● Mutually exclusive (disjoint): refers to two (or more) events that cannot both occur when the random experiment is formed.
A∩B=Ø
● Exhaustive: refers to event(s) that comprise the sample space.
A∪B=𝕌
● Partition: events that are both mutually exclusive and exhaustive.
A∩B=Ø and A∪B=𝕌
⛲ Example 1. All the clubs are taken out of a pack of cards. The remaining cards are then shuffled and one card chosen. After being chosen, the card is replaced before the next card is chosen.
a) What is the sample space?
Solution:
{deck without clubs}
b) Find a set to represent the event, P, of drawing a picture card.
Solution:
P={J; Q; K of hearts, diamonds or spades}
c) Find a set for the event, N, of drawing a numbered card.
Solution:
N={A; 2; 3; 4; 5; 6; 7; 8; 9; 10 of hearts, diamonds or spades}
d) Represent the above events in a Venn diagram.
Solution:
e) What description of the sets P and N is suitable? (Hint: Find any elements of P in N and of N in P.)
Solution:
Mutually exclusive and complementary/exhaustive.
⛲ Question 1. A die is rolled. Let C be the event “die shows 4” and D be the event “die shows even number”. Are C and D mutually exclusive?
Solution:
When a die is rolled, the sample space is given by
S={1,2,3,4,5,6}
Accordingly, C={4} and D={2,4,6}
It is observed that C∩D={4}≠Ø
Therefore, C and D are not mutually exclusive events.
⛲ Q2. An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: E: the sum is greater than 8, F: 2 occurs on either die, G: the sum is at least 7 and a multiple of 3.
Which pairs of these events are mutually exclusive?
Solution:
When a pair of dice is rolled, the sample space is given by
Hence, events E and F; events F and G are mutually exclusive.
⛲ Q3. Three coins are tossed once. Let H denote the event ‘three heads show”, I denote the event “two heads and one tail show”, J denote the event “three tails show and K denote the event ‘a head shows on the first coin”. Which events are
(i). mutually exclusive? (ii). simple? (iii). Compound?
Solution:
When three coins are tossed, the sample space is given by
(i). Events H and I; events H and J; events I and J; and events J and K, each of those pairs is mutually exclusive.
(ii). If an event has only one sample point of a sample space, it is called a simple event. Thus, H and J are simple events.
(iii). If an event has more than one sample point of a sample space, it is called a compound event. Thus, I and K are compound events.
⛲ Example 2. Three coins are tossed. Describe
(i). Two events which are mutually exclusive.
(ii). Three events which are mutually exclusive and exhaustive.
(iii). Two events, which are not mutually exclusive.
(iv). Two events which are mutually exclusive but not exhaustive.
(v) Three events which are mutually exclusive but not exhaustive.
Answer:
(i). “Getting at least two heads”, and “getting at least two tails”
(ii). “Getting no heads”, “getting exactly one head” and “getting at least two heads”
(iii). “Getting at most two tails”, and “getting exactly two tails”
(iv). “Getting exactly one head” and “getting exactly two heads”
(v). “Getting exactly one tail”, “getting exactly two tails”, and getting exactly three tails”
Note There may be other events also as answer to the above question.
Solution:
When three coins are tossed, the sample space is given by
S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
(i). Two events that are mutually exclusive can be L: getting no heads and M: getting no tails
This is because sets L={TTT}and M={HHH} are disjoint.
(ii). Three events that are mutually exclusive and exhaustive can be N: getting no heads O: getting exactly one head P: getting at least two heads
i.e.,
N={TTT} O={HTT,THT,TTH} P={HHH,HHT,HTH,THH}
This is because N∩O=O∩P=P∩N=Ø and N∪O∪P=S.
(iii). Two events that are not mutually exclusive can be Q: getting three heads R: getting at least 2 heads
i.e.,
Q={HHH} R={HHH,HHT,HTH,THH}
This is because Q∩R={HHH}≠Ø
(iv). Two events which are mutually exclusive but not exhaustive can be T: getting exactly one head V: getting exactly one tail
That is
T={HTT,THT,TTH} V={HHT,HTH,THH}
It is because, T∩V=Ø, but T∪V≠S
(v). Three events that are mutually exclusive but not exhaustive can be W: getting exactly three heads X: getting one head and two tails Y: getting one tail and two heads
i.e.,
