Independent events
In probability and statistics, an independent event is an event that is not affected by the outcome of any other events. This is in contrast to dependent events. One example of an independent event is a coin toss. Assuming that the coin is fair and that it can only land on heads or tails, there is an equal probability (0.5) of either heads or tails occurring with each toss of the coin. It doesn't matter if the previous coin toss resulted in the coin landing on heads. There is still an equal probability of either tails or heads occurring on each subsequent toss of the coin.
Probability of independent events
The probability of an independent event occurring, written as P(A) is calculated as:
Since a fair coin has two sides, heads or tails, there is only one way that heads or tails can occur, and since there are two possible outcomes, the probability of either occurring is equal to ½, or 50%.
Calculating the probability of multiple independent events occurring is relatively simple. Since independent events do not affect each other, the probability of two independent events occurring can be found as the product of their individual probabilities. Using probability notation, the probability of both independent events A and B occurring can be written as:
P(A and B) = P(A) × P(B)
or
P(A ∩ B) = P(A) × P(B)
where "∩" is the intersection symbol, and indicates that both events are occurring.
Example
What is the probability of rolling a 1 on the first roll and a 6 on the second roll of a die?
The probability of getting either a 1 or a 6 on any roll is 1/6 since there is one way for each event to occur out of 6 possible outcomes. The probability of both occurring is the product of their individual probabilities. So, given that P(A) is the probability of rolling a 1 and P(B) is the probability of rolling a 6:
P(A ∩ B) = 1/6 × 1/6 = 1/36 = 0.0278
There is approximately a 2.78% chance of rolling a 1 followed by a 6.