Unlocking the Addition Rule of Probability: A Deep Dive

Event	Favorable Outcomes	Probability
$A$	$1,$ $2$	$\frac{2}{6} = \frac{1}{3}$
$B$	$5,$ $6$	$\frac{2}{6} = \frac{1}{3}$

Event

Favorable Outcomes

Probability

A

1,

2

\frac{2}{6} = \frac{1}{3}

B

5,

6

\frac{2}{6} = \frac{1}{3}

	Likes Classics	Does Not Like Classics
Likes Romances	$8$	$11$
Does Not Like Romances	$20$	$11$

Likes Classics

Does Not Like Classics

Likes Romances

8

11

Does Not Like Romances

20

11

To find the probability that a randomly chosen club member likes classics or romances, it is important to identify the number those who like classics and the number of those who like romances. To do so, a convenient method is to find the marginal frequencies with the aid of a two-way table. The marginal frequencies show the answers of a single question.

	Likes Classics	Does Not Like Classics	Total
Likes Romances	$8$	$11$	$19 8 + 11 =$
Does Not Like Romances	$20$	$11$	$31 20 + 11 =$
Total	$28 8 + 20 =$	$22 11 + 11 =$	$50$

Referencing the table, Ramsha wrote the following data on her notepad.

It is important to notice that if these marginal frequencies are added, the number of club members that like both classics and romances will be counted twice. That create false results. Therefore, it is critical to subtract this number —

8

— from the sum of

28

and

19

to find the actual number of members that like both movie genres.

28 + 19 - 8 = 39

The experimental probability that a randomly chosen members likes either classics or romances is calculated by dividing

39

from the total number of members surveyed.

P (classics or romances) = \frac{3 9}{5 0}

CalcQuot

Calculate quotient

P (classics or romances) = 0.78

It should be noted that this probability — expressed as a fraction — can also be expressed in terms of the probabilities of the different events. That can be done if the combined probability's fraction can be written as a sum of fractions. Take the following interpretation as an example.

P (classics or romances) = \frac{3 9}{5 0}

Rewrite

Rewrite $39$ as $28 + 19 - 8$

P (classics or romances) = \frac{2 8 + 1 9 - 8}{5 0}

WriteSumFrac

Write as a sum of fractions

P (classics or romances) = \frac{2 8}{5 0} + \frac{1 9}{5 0} - \frac{8}{5 0}

These fractions are the probabilities that a randomly chosen cinema club member likes classics, romances, or both, respectively.

P (classics or romances) = \frac{2 8}{5 0} + \frac{1 9}{5 0} - \frac{8}{5 0}

SubstituteValues

Substitute values

P (classics or romances) = P (classics) + P (romances) - P (classics and romances)

Well, it looks like for their movie night people might be clamoring for either a classic or a romance! Ramsha decides this is cool with her, and she'll join after all.

Event	Outcome(s)	Probability
Even	$2,$ $4,$ $6$	$P (A) = \frac{3}{6} = \frac{1}{2}$
Prime	$2,$ $3,$ $5$	$P (B) = \frac{3}{6} = \frac{1}{2}$
Even and prime	$2$	$P (A and B) = \frac{1}{6}$

Event

Outcome(s)

Probability

Even

2,

4,

6

P (A) = \frac{3}{6} = \frac{1}{2}

Prime

2,

3,

5

P (B) = \frac{3}{6} = \frac{1}{2}

Even and prime

2

P (A and B) = \frac{1}{6}

Notation	Meaning
$P (A) = a + c$	The probability of $A$ happening is $a + c .$
$P (B) = b + c$	The probability of $B$ happening is $b + c .$
$P (A \cup B) = a + b + c$	The probability of $A$ happening or $B$ happening is $a + b + c .$

Notation

Meaning

P (A) = a + c

The probability of

A

happening is

a + c .

P (B) = b + c

The probability of

B

happening is

b + c .

P (A \cup B) = a + b + c

The probability of

A

happening or

B

happening is

a + b + c .

Event	Favorable Outcomes	Probability
$A$	$2 ♣,$ $4 ♣,$ $6 ♣,$ $8 ♣,$ $10 ♣,$ $2 ♠,$ $4 ♠,$ $6 ♠,$ $8 ♠,$ $10 ♠,$ $2 ♢,$ $4 ♢,$ $6 ♢,$ $8 ♢,$ $10 ♢,$ $2 ♡,$ $4 ♡,$ $6 ♡,$ $8 ♡,$ $10 ♡$	$\frac{2 0}{5 2}$
$B$	$A ♣,$ $2 ♣,$ $3 ♣,$ $4 ♣,$ $A ♠,$ $2 ♠,$ $3 ♠,$ $4 ♠,$ $A ♢,$ $2 ♢,$ $3 ♢,$ $4 ♢,$ $A ♡,$ $2 ♡,$ $3 ♡,$ $4 ♡$	$\frac{1 6}{5 2}$
$C$	$A ♡,$ $2 ♡,$ $3 ♡,$ $4 ♡,$ $5 ♡,$ $6 ♡,$ $7 ♡,$ $8 ♡,$ $9 ♡,$ $10 ♡,$ $J ♡,$ $Q ♡,$ $K ♡$	$\frac{1 3}{5 2}$
$A and B$	$2 ♣,$ $4 ♣,$ $2 ♠,$ $4 ♠,$ $2 ♢,$ $4 ♢,$ $2 ♡,$ $4 ♡$	$\frac{8}{5 2}$
$A and C$	$2 ♡,$ $4 ♡,$ $6 ♡,$ $8 ♡,$ $10 ♡$	$\frac{5}{5 2}$
$B and C$	$A ♡,$ $2 ♡,$ $3 ♡,$ $4 ♡$	$\frac{4}{5 2}$
$A and B and C$	$2 ♡,$ $4 ♡$	$\frac{2}{5 2}$

Event

Favorable Outcomes

Probability

A

2 ♣,

4 ♣,

6 ♣,

8 ♣,

10 ♣,

2 ♠,

4 ♠,

6 ♠,

8 ♠,

10 ♠,

2 ♢,

4 ♢,

6 ♢,

8 ♢,

10 ♢,

2 ♡,

4 ♡,

6 ♡,

8 ♡,

10 ♡

\frac{2 0}{5 2}

B

A ♣,

2 ♣,

3 ♣,

4 ♣,

A ♠,

2 ♠,

3 ♠,

4 ♠,

A ♢,

2 ♢,

3 ♢,

4 ♢,

A ♡,

2 ♡,

3 ♡,

4 ♡

\frac{1 6}{5 2}

C

A ♡,

2 ♡,

3 ♡,

4 ♡,

5 ♡,

6 ♡,

7 ♡,

8 ♡,

9 ♡,

10 ♡,

J ♡,

Q ♡,

K ♡

\frac{1 3}{5 2}

A and B

2 ♣,

4 ♣,

2 ♠,

4 ♠,

2 ♢,

4 ♢,

2 ♡,

4 ♡

\frac{8}{5 2}

A and C

2 ♡,

4 ♡,

6 ♡,

8 ♡,

10 ♡

\frac{5}{5 2}

B and C

A ♡,

2 ♡,

3 ♡,

4 ♡

\frac{4}{5 2}

A and B and C

2 ♡,

4 ♡

\frac{2}{5 2}

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Catch-Up and Review

Challenge

Calculating the Chances of Winning a Card Game

Example

Finding the Probability of a Compound Event When Rolling a Die

Hint

Solution

Example

A Survey, Probabilities, and a Decision

Hint

Solution

Discussion

Addition Rule of Probability

Proof

Example

Calculating the Probability of the Union of Two Events

Hint

Solution

Pop Quiz

Practice Calculating Probabilities of the Union of Two Events

Example

Calculating the Probability of the Union of Three Overlapping Events

Answer

Hint

Solution

Closure

Probability of Winning a Card Game with Multiple Winning Conditions

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