# Independent Events

In these lessons, we will learn how to find the probability of independent events. We will also learn the difference between the probability of dependent events and the probability of independent events.

### What are Independent Events in Probability?

Events are independent if the outcome of one event does not affect the outcome of another. For example, if you throw a die and a coin, the number on the die does not affect the result you get on the coin.

How to calculate the probability of independent events?
If A and B are independent events, then the probability of A happening AND the probability of B happening is P(A) × P(B).

The following gives the multiplication rule to find the probability of independent events occurring together. Scroll down the page for more examples and solutions of word problems that involve the probability of independent events. Example:
If a dice is thrown twice, find the probability of getting two 5’s.

Solution:   Example:
Two sets of cards with a letter on each card as follows are placed into separate bags. Sara randomly picked one card from each bag. Find the probability that:
a) She picked the letters ‘J’ and ‘R’.
b) Both letters are ‘L’.
c) Both letters are vowels.

Solution:
a) Probability that she picked J and R = b) Probability that both letters are L = c) Probability that both letters are vowels = Example:
Two fair dice, one colored white and one colored red, are thrown. Find the probability that:
a) the score on the red die is 2 and white die is 5.
b) the score on the white die is 1 and red die is even.

Solution:
a) Probability the red die shows 2 and white die 5 = b) Probability the white die shows 1 and red die shows an even number = How to determine the probability of independent events
The probability of an event represents the likelihood it will occur. Probability compares the favorable number of outcomes to the total number of outcomes.

Probability can be expressed as a fraction, decimal, or percentage.

The total number of outcomes is often called the sample space.

The favorable number of outcomes is often called the event.

Events A and B are independent events if the probability of Event B occurring is the same whether or not Event A occurs.

Examples of independent events:

1. A fair die is rolled two times.
2. Picking a card from a deck and flipping a fair coin.

Example of dependent events:

1. Draw one card from a deck without replacement and then draw another card.
When two events are independent, the probability of both occurring is the product of the probabilities of the individual events.
P(A and B) for independent events
If events A and B are independent, then the probability of both A and B occurring is
P(A and B) = P(A) ˙ P(B)
where P(A and B) is the probability of events A and B both occurring, P(A) is the probability of event A occurring, and P(B) is the probability of event B occurring.

Examples:

1. You flip a fair coin and pick 1 card out of a hat containing 20 cards numbered 1 - 20. What is the probability of getting heads on the coin and a number greater than 15 form the hat?

2. You roll a fair die twice. What is the probability of rolling a 3 on the first roll and an even number on the second roll?

3. A cart is pulled from a deck of cards and noted. The card is then replaced, the deck is shuffled, and a second card is removed and noted. What is the probability that both cards are face cards?

How to calculate probability of AND statements of independent events?
Two events are independent if the outcome of one event does not affect the likelihood of the other event.

Let A and B be independent events. Then the probability of A and B occurring is:
P(A and B) = P(A ∩ B) = P(A) ˙ P(B)

Example:
P(Flipping heads and rolling a 5 on a 6-sided dice)

Examples of calculating the Probability of Independent Events

Example:
Two coins are tossed. Find the probability of the following event.

### Compare Dependent And Independent Events

Statistics - Dependent and Independent Events
This lesson teaches the distinction between Independent and Dependent Events, and how to calculate the probability of each.

The probability of two events is independent if what happens in the first event does not affect the probability of the second event. P(A + B) = P(A) × P(B)

The probability of two events is dependent if what happens in the first event does affect the probability the second event. P(A + B) = P(A) &times P(B after A)

Example 1: If I roll a pair of dice, what is the probability that both dice land on a 6? Are these dependent or independent events?

Example 2: There are 4 puppies; two are male and two are female. If you randomly pick two puppies, what is the probability that they will both be female? Are these Independent or Dependent Events?

Example 3: You randomly choose one of the letter cubes. Without replacing it, you now choose a 2nd letter cube and place it to the right of the 1st letter cube. Then you pick a 3rd letter cube, and place it to the right of the 2nd letter cube. What is the probability that the letter cubes now spell “BIT”?

Independent vs Dependent Probability

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 