Probability - P(A or B)

Definition

P(A or B) simply means "The probability of occurrence of event A or event B".

For example, suppose event A is "getting a 4 on rolling a number cube" and event B is "getting heads on tossing a coin", then P(A or B) represents the probability of either of the following three situations:

• You do get 4 on rolling the number cube and you don't get heads on tossing the coin
• You don't get 4 on rolling the number cube but you do get heads on tossing the coin
• You get both 4 on rolling the number cube and heads on tossing the coin

Note that if you don't get 4 on a rolling the number cube and you don't get heads on tossing the coin, then we say that "A or B" has not occurred.

Thus, if either event A occurs or event B occurs, or both the events A and B occur together, then we say that "A or B" has occurred.

P(A or B) is also represented by P(A U B) and P(A + B).

Understand

All the three above expressions, P(A U B), P(A or B) and P(A + B) mean the same thing, "probability of A or B".

For example, you roll a number cube. Then the expression P(1 or 6) means "probability of getting 1 or 6 on rolling the number cube".

How to Calculate

The probability of P(A or B) is calculated either by manually listing the elements in set `AUB` or by the Addition Theorem. Both of these methods are discussed with a sample problem below.

Formula

The addition theorem of probability can be used to calculate P(A or B). It is as follows:

`P("A or B") = P(A) + P(B) - P("A and B")`

Although you can use the above formula to calculate P(A or B), but, as you will see in the examples below, most questions are solved by simply counting the number of events in the set "A or B".

The following example illustrates both methods of calculating P(A or B).

Problem

Calculate the probability P(Even or greater than 4) on rolling a number cube.

Method 1: Manually list the elements in set AUB

The question is asking us to determine the probability of getting a number which is either even or prime on rolling a number cube.

In this method we will list the elements in set AUB (A union B). The given probability is P(Even or 3). Thus we will use the elements of the set "Even U Prime".

Even numbers on a number cube are {2, 4, 6} and the numbers greater than 4 are {5, 6}. Let A = {2, 4, 6} and B = {5, 6}. Then,

AUB = {2, 4, 5, 6}

There are four elements in set AUB. Thus number of favorable outcomes = 4. (these are the outcomes which favor the probability of either an even number or a prime number on rolling the number cube).

We already know that there are six faces on a number cube and hence total number of possible outcomes is 6. Thus, by the definition of theoretical probability,

P(Even or Prime) = `"Number of favorable outcomes"/"Total number of outcomes"`

P(even or prime) = `4/6 = 2/3`

Thus we found that the probability of getting a number which is either even or a prime number is `2/3`.

Method 2: Addition Theorem

The addition theorem of probability helps you calculate the probability of event A or event B. It is as follows.

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

Thus in the above question, we can write,

`P("Even or Prime") = P("Even") + P("prime") - P("even and prime")`

Thus, we calculate each probability P(even), P(prime) and P(even and prime) individually and then plug them in the formula above.

`P("even") = "Number of even numbers on a number cube"/"Total number of numbers on a number cube" = 3/6 = 1/2`

(There are three even numbers on a number cube: 2, 4 and 6.)

`P("prime") = "Number of prime numbers on a number cube"/"Total number of numbers on a number cube" = 4/6 = 2/3`

(There are four prime numbers on a number cube: 1, 2, 3 and 5.)

`P("even and prime") = "Number of numbers on a number cube which are both even and prime"/"Total number of numbers on a number cube" = 1/6`

(2 is the only number which is prime and even.)

Now we plug in the above probability values into the addition theorem.

`P("even or prime") = 1/2 + 2/3 - 1/6`

`P("even or prime") = (3 + 2 - 1)/6 = 4/6 = 2/3`

Thus we calculated the same probability by using the addition theorem.

Some more solved examples are provided below.

Solved Examples

Problem 1

Calculate the probability of getting either an even number or 3 on rolling a number cube.

To solve this problem, first we will find the number of elements in the set "even number or 3". There are six faces on a number cube, numbered 1 to 6. Out of these, three are even: 2, 4 and 6. Thus, including 3, there are 4 faces. Thus, the number of favorable outcomes is 4.

Further, since any of the six faces may show up on rolling the number cube, therefore the total number of possible outcomes is 6.

Applying the formula for theoretical probability,

`P(A) = "Number of favorable outcomes"/"Total number of outcomes"`

`P(even or 3) = 4/6 = 2/3`

Problem 2

There are ten marbles in a bag: 2 red, 3 green, 4 blue and 1 yellow. You pick one marble out of the bag. Find the probability that the marble will be either blue or yellow.

Total number of marbles in the bag = 10. Thus total number of possible outcomes is 10. Further, there are 4 blue and 1 yellow marbles in the bag. Thus, number of favorable outcomes is 4 + 1 = 5.

Applying the formula for theoretical probability,

`P(A) = "Number of favorable outcomes"/"Total number of outcomes"`

`P(blue or yellow) = 5/10 = 1/2`

Problem 3

What is the probability of getting either heads or tails on tossing a coin?

This question is very easy to answer - the probability of getting either heads or tails on tossing a coin is 1. This is because you can get either heads or tails on tossing a coin and there is no other possible outcome. Thus, the probability that you get either heads or tails is 1.

Problem 4

On rolling two number cubes, find the probability that the sum of the numbers you get on both the cubes is either 10 or 12.

First, we list all the possible combinations of numbers on the two number cubes that give a sum of 10:

• 4 + 6 = 10
• 5 + 5 = 10
• 6 + 4 = 10

Now, we list all the possible combinations of numbers which give a sum of 12

• 6 + 6 = 12

Now, we count the number of combinations which give either sum of 10 or a sum of 12. Three combinations give a sum of 10 and one gives a sum of 12. Thus a total of 4 combinations give a sum of 10 or 12. Therefore the number of favorable outcomes is 4.

Now, we calculate the total number of possible outcomes. When rolling a cube, we can get either one of the six faces on it. Thus, there are six possible outcomes on rolling one number cube. When we roll two number cubes, the number of possible outcomes on the second cube for each outcome of the first one are 6. Thus, in total there are (by applying the fundamental principle of counting) `6 times 6 = 36` possible outcomes on rolling two number cubes.

Applying the formula for theoretical probability,

`P(A) = "Number of favorable outcomes"/"Total number of outcomes"`

`P(blue or yellow) = 4/36 = 1/9`

Problem 5

There are three red, four blue and three white marbles in a bag. One marble is selected from the bag. What is the probability that it will be either red or blue?

Total number of marbles in the bag are 3 + 4 + 3 = 10. Thus the total number of possible outcomes is 10. Number of marbles that are either red or blue are 3 + 4 = 7. Thus, the number of favorable outcomes is 7.

Applying the formula for theoretical probability,

`P(A) = "Number of favorable outcomes"/"Total number of outcomes"`

`P(blue or yellow) = 7/10`

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