In these lessons, we will learn how to find the probability of mutually exclusive events. We will also compare mutually exclusive events and independent events.

**Related Pages**

More Lessons On Probability

Probability Tree Diagrams

Independent Events

Dependent Events

The following diagrams show the formulas for the probability of mutually exclusive events and non-mutually exclusive events. Scroll down the page for examples and solutions.

Two events are said to be **mutually exclusive** if they cannot
happen at the same time.

For example, if we toss a coin, either heads or tails might turn up, but not heads and tails at the same time. Similarly, in a single throw of a die, we can only have one number shown at the top face. The numbers on the face are mutually exclusive events.

If *A* and *B* are mutually exclusive events then the probability of *A* happening
**OR** the probability of *B* happening is P(*A*) + P(*B*).

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

**Example:**

The probabilities of three teams A, B and C winning a badminton competition
are 1/3, 1/5 and 1/9 respectively.

Calculate the probability that

a) either A or B will win

b) either A or B or C will win

c) none of these teams will win

d) neither A nor B will win

**Solution:**

c) P(none will win) = 1 – P(A or B or C will win)

d) P(neither A nor B will win) = 1 – P(either A or B will win)

Probabilities of Mutually Exclusive Events

If two events are ‘mutually exclusive’ they cannot occur at the same time.

Learn all about mutually exclusive events in this video.

For mutually exclusive events the total probabilities must add up to 1.

Probability - P(A ∪ B) and Mutually Exclusive Events

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

For mutually exclusive events, P(A ∩ B) = 0.

The following video shows how to calculate the probability of mutually exclusive events and non-mutually exclusive events.

**Examples:**

- Find the probability of drawing a yellow ball or drawing a three.
- Find the probability of drawing a red ball or drawing an odd number ball.

**Examples:**

- What is the probability of drawing a heart and a black card?
- What is the probability of drawing a heart or a black card?
- What is the probability of drawing a heart and a face card?
- What is the probability of drawing a heart or a face card?

**Example:**

The figure shows how 25 people travelled to work: B for bicycle, T for Train and W for Walk.

a) Write down two of these events that are mutually exclusive. Give a reason for your answer.

b) Determine whether or not B and T are independent events.

Mutually exclusive events cannot be independent events.

Independent events cannot be mutually exclusive events.

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