4.5 Independent Events

Two trials (or events or results of a random experiment) are independent trials if the outcome of one trial does not influence the outcome of the second trial. If two events are not independent, they are dependent events. For instance, if two coins are flipped they are independent since flipping one coin does not affect the outcome of the second coin.

Be careful with this rule. You cannot just multiply probabilities to find an intersection unless you know they are independent. Also, do not confuse independent events with mutually exclusive events. Two events are mutually exclusive when the P(A ∩ B) = 0.

Example 3

The probability that a student has their own laptop is 0.78. If three students are randomly selected, what is the probability that at least one owns a laptop?

Show solution

There is an assumption that the three students are not related and that the probability of one owning a laptop is independent of the other people owning a laptop. The probability of none owning a laptop is (1 – 0.78)3 = 0.0106. The probability of at least one is the same as 1 – P(None) = 1 – 0.0106 = 0.9894.

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University), hosted on LibreTexts (stats.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.