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Applications of Probability

Addition Rule in the Uniform Probability Model

This lesson breaks down the calculation of the probability of real-world compound events using the Addition Rule of Probability.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Calculating the Chances of Winning a Card Game

Diego is playing cards using a standard deck.

Diego wins if he draws a card that satisfies any of the following events.
If Diego draws a card from a full deck, what is the probability that he wins? Tell Diego his chances of winning by rounding to one decimal place.

Example

Finding the Probability of a Compound Event When Rolling a Die

Consider the experiment of rolling a regular six-sided die.
Die
The events A and B are defined as follows.
These events can be illustrated using a Venn Diagram.
Venn Diagram
It is also possible to consider the event that results from the union of A and B, written as A or B. Find the probability that the event A or B occurs. Round the probability to one decimal place.

Hint

Which are the favorable outcomes for the event A or B?

Solution

To find the probability of these events, the number of favorable outcomes of each event is divided by the total number of outcomes. Since it is known that there are six possible outcomes for rolling a die, the probabilities can be found by identifying the favorable outcomes for the events.

Event Favorable Outcomes Probability
A 1, 2
B 5, 6

A diagram can be used to represent the favorable outcomes for the event A or B — the outcomes that satisfy either A or B.

Venn Diagram showing the union of the events
Since A and B do not share any favorable outcomes, these events are not overlapping events. As can be seen, there are two favorable outcomes for each event. Therefore the total number of favorable outcomes for event A or B is 4. To find the probability that the event A or B occurs, the number of favorable outcomes of this event is divided by the number of possible outcomes.
It is important to understand that the probability of the event A or B is equal to the sum of the probabilities of the events A and B.
As shown in the previous situation, the compound probability was found for events that did not overlap. When events overlap, the situation needs to be approached in a slightly different way.

Example

A Survey, Probabilities, and a Decision

Ramsha took a survey of her school's cinema club members just to get a sense of their taste before she considers joining. She asked 50 members two questions about movie genres: First, do they like classics? And second, do they like romances? The results are displayed in a two-way frequency table.

Likes Classics Does Not Like Classics
Likes Romances 8 11
Does Not Like Romances 20 11
What is the experimental probability that a randomly chosen member likes classics or romances? Round the probability to two decimal places.

Hint

What can be done about the value for members that like both classics and romances?

Solution

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
Does Not Like Romances 20 11
Total 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.
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)=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.
These fractions are the probabilities that a randomly chosen cinema club member likes classics, romances, or both, respectively.
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.
Next, the general formula for finding the probability of the union of two events will be developed.

Discussion

Addition Rule of Probability

For two mutually exclusive events A and B, the probability that A or B occur in one trial is the sum of the individual probability of each event.

P(A or B)=P(A)+P(B)
For example, consider rolling a standard six-sided die. Let A be the event that a 3 is rolled and B be the event that a 4 is rolled. The probability of A or B can be found by adding the individual probabilities.
The formula above can be generalized to events that are not necessarily mutually exclusive. If events are overlapping, the probability of the common outcomes are counted twice in P(A)+P(B), so an adjustment is needed.
P(A or B)=P(A)+P(B)P(A and B)

For example, consider rolling a standard six-sided die. Let A be the event that an even number is rolled and B be the event that a prime number is rolled.

Event Outcome(s) Probability
Even 2, 4, 6
Prime 2, 3, 5
Even and prime 2
Using the formula gives the probability that the result of the roll is even or prime.
This probability can be verified by accounting for the five outcomes that are even or prime: 2, 3, 4, 5, and 6.

Proof

Proving the Addition Rule of Probability

For mutually exclusive events, the Addition Rule of Probability is a postulate.

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

Therefore, no proof will be given for mutually exclusive events. Now, consider non-mutually exclusive events A and B.

Venn diagram showing two overlapping sets
In the Venn diagram above, it can be seen part of event A does not overlap event B. That part is labeled a. Similarly, the part of event B that does not overlap event A is labeled b. Furthermore, the overlapping part – also known as the intersection — of both events is labeled c.
Furthermore, in the diagram it can be also seen that a, b, and c are mutually exclusive. Therefore, the union of event A and event B should be considered.
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.
The probability of A happening or B happening is a+b+c.
Finally, the fact that will be used to prove the Addition Rule of Probability for non-mutually exclusive events.

Rewrite 0 as cc

The rule has been proven for non-mutually exclusive events.
Remember that if two events A and B are mutually exclusive, then P(A and B)=0. Therefore, the second formula presented can always be used regardless of the events being mutually exclusive or not.

Example

Calculating the Probability of the Union of Two Events

Consider the following probabilities for events A and B.

  • P(A)=0.13
  • P(B)=0.37
  • P(A and B)=0.09
What is the value of P(A or B)?

Hint

Use the formula for the Addition Rule of Probability.

Solution

The Addition Rule of Probability can be used to find the value of P(A or B).
The given values can be substituted in the formula to find the desired probability.
P(A or B)=P(A)+P(B)P(A and B)
P(A or B)=0.13+0.370.09
P(A or B)=0.41

Pop Quiz

Practice Calculating Probabilities of the Union of Two Events

Use the Addition Rule of Probability to answer each question. Write each probability rounded to two decimal places.

Addition Rule of Probability Randomized
Now that the skill of finding the probability of the union of two events has been accomplished, how about finding the probability of the union of three events?

Example

Calculating the Probability of the Union of Three Overlapping Events

As shown in the following diagram, A, B, and C are three overlapping events.

Three overlapping events: A, B, and C

Write an expression for P(A or B or C).

Answer

P(A or B or C) = P(A) + P(B) + P(C)P(A and B) P(B and C) P(A and C) + P(A and B and C)

Hint

Use a Venn diagram to construct the expression.

Solution

To find an expression for this probability, it is convenient to start with the known Addition Rule of Probability for two overlapping events.
This expression can be extended for three events by adding the probabilities of the three events and subtracting the intersections of two of those events.
That expression might appear a bit muddled. To illustrate what happens when these quantities are added and subtracted, a Venn diagram can be used. The numbers on each section illustrate how many times the probability associated with that section has been added to the total.
probability of three overlapping events first try
It can be seen that P(A and B and C) is not added in the total. Therefore, it is needed to add this probability to complete the expression.
This expression can be illustrated using a Venn diagram.
probability of three overlapping events

Closure

Probability of Winning a Card Game with Multiple Winning Conditions

The challenge presented at the beginning of this lesson asked for the probability that Diego draws a card that satisfies either of these events.
It can be seen that these three events are overlapping events because there are cards that satisfy more than one event at a time. Previously, an expression for P(A or B or C) was found. This expression can be used to find the probability that Diego wins.
A letter can be associated to each of the events described.
A standard deck of cards has 52 cards. Knowing this, it is possible to find the probability of each event — including compound events — by considering the favorable outcomes. Note that aces will be considered as a number 1, while face cards will not be assigned a number value in this game.
Event Favorable Outcomes Probability
A
B
C
A and B
A and C
B and C
A and B and C
Now the probabilities from the table can be substituted into the expression for P(A or B or C).
Evaluate
Therefore, the probability that Diego wins is about 0.7.
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