Probability – Mutually Exclusive Events or Not

Mutually Exclusive Events

Mutually exclusive events are events, which cannot be true at the same time.
Examples of mutually exclusive events are:
1. A die landing on an even number or landing on an odd number.
2. A student passing or failing an exam
3. A tossed coin landing on heads or landing on tails

This means that if we examine the elements of the sets that make up A and B there will be no elements in common. Therefore, A∩B=∅ (where ∅ refers to the empty set). Since, P(A∩B)=0, equation

P(A∪B)=P(A)+P(B)-P(A∩B)

becomes:
P(A∪B)=P(A)+P(B)

For mutually exclusive events.

For example, suppose a container contains 7 red marbles and 6 blue marbles. The collection of red marbles is a subset of all the marbles. The same is true for the collection of blue marbles.
● Number of marbles in the sample space

=n(S)=13

● Number of marbles in the red subset =n(R)=7
● Number of marbles in the blue subset =n(B)=6
● thus: n(S)=n(R)+n(B)

Mutually exclusive events

Mutually exclusive events are events that cannot happen at the same time. There is no intersection between the events.
Mutual: applies to two or more people or events.
Exclude: to keep out, not allow a person in.
Mutually exclusive: Both events keep the other out. So there is no outcome that can happen in both events at the same time.

Example 1:
If you roll a die, it is impossible for it to land on a l and a 6 at the same time. So P(l) and P(6) are mutually exclusive. When you roll a die, what are the chances of getting a 6 or a 1?
So P(1 or 6)=P(1)+P(6)=⅙+⅙=2/6=⅓
So the chance of rolling either a l or a 6 is ⅓ or 33.3%

S: Possible outcomes for rolling a die


When two events are mutually exclusive, P(A and B)=0
∴P(A or B)=P(A)+P(B) for mutually exclusive events
We can also use this rule for the number of elements or outcomes in each event, if the events are mutually exclusive:
n(A or B)=n(A)+n(B)

When the two events are mutually exclusive, then they do not overlap. Therefore the intersection of A and B is empty and we write A∩B=∅ (empty set) and P(A∪B)=0.
Hint:
If P(A and B)=0 or if P(A or B)=P(A)+P(B), then the events are mutually exclusive.

State whether the following events are mutually exclusive or not.

1. A fridge contains orange juice, apple juice and grape juice. A cooldrink is chosen at random from the fridge. Event A: the cooldrink is orange juice. Event B: the cooldrink is apple juice.
solution:
We are choosing just one cooldrink from the fridge. This cooldrink cannot be both an orange juice and an apple juice. Therefore these two events are mutually exclusive.

2. A packet of cupcakes contains chocolate cupcakes, vanilla cupcakes and red velvet cupcakes. A cupcake is chosen at random from the packet. Event A: the cupcake is red velvet. Event B: the cupcake is vanilla.
solution:
We are choosing just one cupcake from the packet. This cupcake cannot be both a red velvet cupcake and a vanilla one. Therefore these two events are mutually exclusive.

3. A card is chosen at random from a deck of cards. Event A: the card is a red card. Event B: the card is a picture card.
solution:
We are choosing just one card from the deck. This card can be both a red card and a picture card. Therefore these two events are not mutually exclusive.

4. A cricket team plays a game. Event A: they win the game. Event B: they lose the game.
solution:
The cricket team can either win the game or lose the game. They cannot simultaneously win and lose the game. Therefore these two events are mutually exclusive.
Note that a tie game does not count as either a win or a loss. In a tie neither team can be said to have won the match.

Mutually exclusive events

Shakespeare’s phrase “To be, or not to be: that is the question” is an example of two mutually exclusive events.
Two events are mutually exclusive if they cannot occur at the same time. For n mutually exclusive events the probability is the sum of all probabilities of events:
P=p1+p2+⋯+p(n-1)+pn

Or
P(A or B)=P(A)+P(B)

A and B denotes mutually exclusive events

Example 2. If Jessica rolls a die, what is the probability of getting at least a “3”? solution: There are 4 outcomes that satisfy our condition (at least 3): {3, 4, 5, 6}. The probability of each outcome is ⅙. The probability of getting at least a “3” is:
P=⅙+⅙+⅙+⅙=⅔

Q1. In a committee meeting, there were three freshmen, two sophomores, fivejuniors, and three seniors. If a student is selected at random to be the chairperson, find the probability that the chairperson is a sophomore or junior.
solution:
There are two sophomores and five juniors and a total of 13 students.
P(sophomore or junior)=P(sophomore)+P(junior)

Q2. A box contains 10 red, 30 white, and 20 black marbles. A marble is drawn at random. Find the probability that it is either red or white.
solution:
Number of red marbles =10. Number of while marbles =30. Number of black marbles =20

n(S)=60

Red: P(A)=10/60=⅙, white P(B)=30/60=½
P(A∪B)=P(A)+P(B)=⅙+½=⅔

Q3. A dice is thrown twice. What is the probability that sum of the number of dots appearing on them are 3 or 11?
solution:
When two dice are rolled then the number of possible outcomes is n(S)=36.
Let A be the event the sum is 3, then •


Let B be the event that the sum is 11, then

Axiomatic Approach to Probability

In earlier sections, we have considered random experiments, sample space and events associated with these experiments. In our day to day life we use many words about the chances of occurrence of events. Probability theory attempts to quantify these chances of occurrence or non occurrence of events.
In earlier classes, we have studied some methods of assigning probability to an event associated with an experiment having known the number of total outcomes.
Axiomatic approach is another way of describing probability of an event. In this approach some axioms or rules are depicted to assign probabilities.

P(A∪B)=P(A)+P(B)-P(A∩B)

The above formula can further be verified by observing this Venn Diagram

If A and B are disjoint sets, i.e., they are mutually exclusive events, then A∩B
Therefore P(A∪B)=P(Ø)=0
Thus, for mutually exclusive events A and B, we have
P(A∪B)=P(A)+P(B)

Probability of mutually exclusive events:
P(A∪B)=P(A)+P(B)

Probability of not mutually exclusive events:
P(A∪B)=P(A)+P(B)-P(A∩B)

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