Unit 2 · Topic 2.5 Beta

Mutually Exclusive Events

The joint probability P(A∩B) is the chance both events happen; in a table it is one cell over the grand total.

Practice 4: Interpret Results

Part 1 · Hook

Why this matters

A raffle at a school fair gives one prize to a freshman, one to a sophomore and so on. Can one student win the freshman prize and the senior prize? Of course not. Some events cannot happen together, and spotting them makes "or" questions easy, while missing an overlap makes answers too big.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. One of 40 students is chosen at random; 8 are juniors. What is P(junior)?

0.2
0.8
8
0.25

Show the answer

$\frac{8}{40} = 0.2$ : equally likely students.

Correct: 0.2:
0.8:
8:
0.25:

2. In a two-way table of 200 students, 15 are seniors who play a sport. What is the joint relative frequency?

0.075
0.15
0.3
15

Show the answer

One cell over the grand total: $\frac{15}{200} = 0.075$ .

Correct: 0.075:
0.15:
0.3:
15:

Part 4 · See it

See it first

Two diagrams of events inside a rectangle of all outcomes. Left: events A and B as separate circles that do not touch, so A and B cannot happen together and P(A and B) = 0. Right: circles that overlap; the shaded overlap is the joint event A and B. — Left: events with no outcome in common, so the probability of both is 0. Right: events that overlap; the overlap is the joint event. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Two events share some outcomesthe joint event $A \cap B$ ("A and B") is those shared outcomes, and its probability is above 0
Two events share no outcomesthey are mutually exclusive: $P(A \cap B) = 0$
Events are mutually exclusivethe probability of one or the other is the sum: $P(A) + P(B)$
Events overlap and you add anywaythe shared outcomes are counted twice and the answer is too big

Part 6 · Key ideas

Key ideas

The joint probability $P(A \cap B)$ is the probability that A and B both happen.
Events are mutually exclusive (disjoint) when they cannot happen together: $P(A \cap B) = 0$ .
For mutually exclusive events, P(A or B) = P(A) + P(B). If they overlap, plain adding double counts.
Two events whose probabilities add to more than 1 cannot be mutually exclusive.

Part 7 · Misconception

A common mistake

The wrong idea: Any two events about different things, such as class and sport, are mutually exclusive.

What actually happens: Mutually exclusive means no outcome is in both. A junior can play a sport, so "junior" and "plays a sport" overlap: the joint probability is 0.09.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Students by class and sport

A school has 200 students. The table shows each student's class and whether the student plays on a school sports team. One student is chosen at random (invented data).

Class and sports team for 200 students
Class	Plays a sport	No sport	Total
Freshman	20	30	50
Sophomore	22	28	50
Junior	18	32	50
Senior	15	35	50
Total	75	125	200

1. What is $P(\text{senior} \cap \text{plays a sport})$ , the probability that the student is a senior and plays a sport?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

The joint event is one cell: 15 seniors play a sport. Out of 200 equally likely students, $\frac{15}{200} = 0.075$ .

Answer: 0.075

2. Are the events "the student is a freshman" and "the student is a senior" mutually exclusive?

No: each event has the same probability, 0.25
No: some freshmen and some seniors play a sport
Yes: together they make up the whole school of 200 students
Yes: no student is both a freshman and a senior

Show the answer

No student is in two classes, so the joint event has no outcomes: $P(\text{freshman} \cap \text{senior}) = 0$ .

No: each event has the same probability, 0.25: Equal probabilities have nothing to do with being mutually exclusive.
No: some freshmen and some seniors play a sport: Sharing a third feature does not put a student in both classes.
Yes: together they make up the whole school of 200 students: Together they cover only 100 students; and covering everything is not the test.
Correct: Yes: no student is both a freshman and a senior: Right: they cannot happen together.

3. Are "the student is a junior" and "the student plays a sport" mutually exclusive?

No, because 18 students are juniors on a team, so the joint probability is 0.09
Yes, because more juniors do not play a sport (32) than play one (18)
Yes, because class and sport are two different variables
No, because both events have probability above 0.2

Show the answer

18 students are juniors who play a sport, so $P(\text{junior} \cap \text{plays a sport})$ $= \frac{18}{200}$ $= 0.09$ , not 0. The events can happen together.

Correct: No, because 18 students are juniors on a team, so the joint probability is 0.09: Right: the joint probability is not 0.
Yes, because more juniors do not play a sport (32) than play one (18): Mutually exclusive means no overlap at all, not "mostly separate".
Yes, because class and sport are two different variables: Events on different variables can happen together, as these do.
No, because both events have probability above 0.2: The size of each probability is not the test; the joint probability is.

Diagram

Events for one roll of a die

A fair die is rolled once. Event V: the roll is even (2, 4, 6). Event F: the roll is a 5. Event G: the roll is greater than 3 (4, 5, 6).

4. Which pair of events is mutually exclusive?

V and G
F and G
G and "the roll is 6"
V and F

Show the answer

V is 2, 4, 6 and F is 5: no outcome is in both. V and G share 4 and 6; F and G share 5.

V and G: They share 4 and 6, so they can happen together.
F and G: A 5 is greater than 3, so F and G happen together on a 5.
G and "the roll is 6": A 6 is in G, so they overlap.
Correct: V and F: Right: an even roll can never be a 5.

5. What is $P(V \cap G)$ , the probability that the roll is even and greater than 3?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

The outcomes in both events are 4 and 6. Two of six equally likely outcomes: $\frac{2}{6} \approx 0.333$ .

Answer: 0.3333

6. Event H has probability 0.6 and event K has probability 0.5. Can H and K be mutually exclusive?

Yes: any two events with different probabilities are mutually exclusive
Yes: mutually exclusive events have probabilities that add to more than 1
No: then the chance of one or the other would be 1.1, more than 1
It depends on whether H makes K more likely to happen

Show the answer

If they could not happen together, the probability of H or K would be $0.6 + 0.5 = 1.1$ , more than 1, which is impossible. So H and K must overlap.

Yes: any two events with different probabilities are mutually exclusive: Different probabilities do not separate events.
Yes: mutually exclusive events have probabilities that add to more than 1: Probabilities of mutually exclusive events add to at most 1.
Correct: No: then the chance of one or the other would be 1.1, more than 1: Right: no probability can exceed 1.
It depends on whether H makes K more likely to happen: The numbers alone settle it: 0.6 + 0.5 is more than 1.

Part 9 · Summary

Summary

The joint probability

P(A \cap B)

is the chance both events happen; in a table it is one cell over the grand total. Events are mutually exclusive when that joint probability is 0. Only then can you find P(A or B) by adding P(A) and P(B).

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic