Independence

TL;DR

• Knowing the result of one event does not change the probability of another event.
• Probabilities remain constant across trials when events are independent.
• Independence is important for accurate predictions, calculations, and unbiased experiments.

Definition

Independence in probability refers to the concept that the outcome of one event does not affect the outcome of another event. In other words, the probability of an event occurring remains the same regardless of any other events that may have occurred previously.

Explanation

When two events are independent, information about one event does not alter the likelihood of the other. This means the probability assigned to an outcome stays constant across trials or observations, even after observing prior results. Independence simplifies probability reasoning and supports valid inferences from samples to populations in statistics.

Examples

Coin flipping

One example of independence in probability is flipping a coin. The probability of flipping a heads or tails on any given flip is always $1/2$ regardless of how many times the coin has been flipped before. If a coin has been flipped 10 times and all 10 flips landed on heads, the probability of the next flip landing on tails is still $1/2$. The outcome of previous flips does not affect the probability of the next flip.

Drawing cards from a deck

Another example of independence in probability is drawing cards from a deck. The probability of drawing a specific card from a deck of 52 cards is $1/52$. This probability does not change regardless of how many cards have been drawn previously. For instance, if 5 cards have been drawn and none of them were the Ace of Spades, the probability of drawing the Ace of Spades on the next draw is still $1/52$. The outcome of the previous draws does not affect the probability of the next draw.

Use cases

• Conducting experiments where independent outcomes allow straightforward probability calculations and predictions.
• Making inferences about a population based on a sample, relying on independence to avoid biased estimates.

Notes or pitfalls

• If outcomes are not independent, analyses can become biased and produce skewed results, potentially leading to incorrect conclusions.

Related terms

• Probability
• Statistics
• Probability theory
• Event
• Sample
• Population
• Bias