Independent Events

In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability that the other will occur. In other words, if events $A$ and $B$ are independent, then the chance of $A$ occurring does not affect the chance of $B$ occurring and vice versa. The concept of independence extends to dealing with collections of more than two events.

Two events are independent if any of the following are true:

1. $\displaystyle P(A|B) = P(A)$
2. $\displaystyle P(B|A) = P(B)$
3. $\displaystyle P(A \ \text{and} \ B) = P(A)\cdot P(B)$

To show that two events are independent, you must show only one of the conditions listed above. If any one of these conditions is true, then all of them are true.

Translating the symbols into words, the first two mathematical statements listed above say that the probability for the event with the condition is the same as the probability for the event without the condition. For independent events, the condition does not change the probability for the event. The third statement says that the probability of both independent events $A$ and $B$ occurring is the same as the probability of $A$ occurring, multiplied by the probability of $B$ occurring.

As an example, imagine you select two cards consecutively from a complete deck of playing cards. The two selections are not independent. The result of the first selection changes the remaining deck and affects the probabilities for the second selection. This is referred to as selecting "without replacement" because the first card has not been replaced into the deck before the second card is selected.

However, suppose you were to select two cards "with replacement" by returning your first card to the deck and shuffling the deck before selecting the second card. Because the deck of cards is complete for both selections, the first selection does not affect the probability of the second selection. When selecting cards with replacement, the selections are independent.

Independent Events

Selecting two cards from a deck by first selecting one, then replacing it in the deck before selecting a second is an example of independent events.

Consider a fair die role, which provides another example of independent events. If a person roles two die, the outcome of the first roll does not change the probability for the outcome of the second roll.

Example

Two friends are playing billiards, and decide to flip a coin to determine who will play first during each round. For the first two rounds, the coin lands on heads. They decide to play a third round, and flip the coin again. What is the probability that the coin will land on heads again?

First, note that each coin flip is an independent event. The side that a coin lands on does not depend on what occurred previously. 

For any coin flip, there is a ${\frac{1}{2}}$ chance that the coin will land on heads. Thus, the probability that the coin will land on heads during the third round is ${\frac{1}{2}}$.

Example

When flipping a coin, what is the probability of getting tails $5$ times in a row?

Recall that each coin flip is independent, and the probability of getting tails is ${\frac{1}{2}}$ for any flip. Also recall that the following statement holds true for any two independent events A and B:

$\displaystyle P(A \ \text{and} \ B) = P(A)\cdot P(B)$

Finally, the concept of independence extends to collections of more than $2$ events.

Therefore, the probability of getting tails $4$ times in a row is:

$\displaystyle {\frac{1}{2}} \cdot {\frac{1}{2}} \cdot {\frac{1}{2}} \cdot {\frac{1}{2}} = {\frac{1}{16}} $