It’s raining and my numbers just came up in the lottery. Obviously, these two events have nothing to do with each other. The probability of my numbers coming up in the lottery is independent from the probability that it rains. These are two independent events.

When the outcome of one event does not influence the outcome of the other event, the two events are independent. Event A does not influence the probability of event B. 

Pick’n roll example

Say you were to pick ace of spade from a standard deck of card and roll a 3 on with a six-sided die. These two events do not influence each other. They are independent events as you would have the same chance of picking the ace of spades whether you roll the die before or after picking the card.

Independent events vs dependent events

On the first draw from a standard deck of 52 card you have a probability of 4 out of 52 of drawing an ace. Say you draw an ace on the first draw. Now, the probability of drawing an ace on the second draw is affected by the first draw, as there are now only three aces remaining in the deck that now only has 51 cards. The probability is now 3 out of 51.

Dependent events do influence each other where independent events do not.

With replacement

In case you put the ace back in the deck of cards after drawing it, you have the same probability of drawing it on the second draw. There are again 4 aces and 52 cards. In this case, the two events are independent. The act of putting back the card is denominated as an experiment “with replacement”.

In The binomial distribution, I go closer on experiments with replacement.

Independent events vs mutually exclusive events

It rains, and my numbers come up in the lottery. These two events can happen at the same time although they have nothing to do with each other. They are independent events but can take place at the same time.

On a flip with a coin I cannot flip a head and a tail at the same time. These two events are not possible at the same time and are therefore mutually exclusive. So, independent events can happen at the same time, where mutually exclusive events cannot.

Calculating rules for independent events

If you flip a coin twice, what is the probability of getting head on first flip and tail on second flip? This is calculated as the product of these two probabilities:

P(A) × P(B) = 0.5  × 0.5 = 0.25

And since, event B does not depend on event A, the probability of event B occurring given that event A has happened does not affect the probability of event B. In other words, that my numbers coming up in the lottery has nothing to do with the fact that it is raining. This can be written:

Venn diagrams and independent events

Venn diagrams have little effect when it comes to illustrating independent events. In a Venn diagram independent events can only be identified when:

Learning more on independent events

• Khan Academy list of tutorial videos on Probability and combinatorics
• Khan Academy video: Independent & dependent probabilities
• Jbstatistics video: What does independence look like on a Venn diagram?