How can I tell whether two probability events are independent?

First, define exactly what A and B mean, then look at how the model treats the chance of one event after the other is known. Independence means that knowing B occurred does not change the probability of A. If P(B) is greater than zero, test whether P(A | B) = P(A). You can also use the equivalent check P(A and B) = P(A)P(B). [S1]

For example, suppose a problem describes two draws from a bag. Before deciding, check whether the first item is returned before the second draw. That rule can affect the probabilities; simply calling the draws separate events cannot settle the question.

Assume a bag contains 2 red and 2 blue balls, and two balls are drawn in order. Let A mean “the first ball is red” and B mean “the second ball is red.” These are illustrative assumptions for the example.

Without replacement, P(A) = 2/4 = 1/2. If the first ball was red, only 1 red ball remains among 3 balls, so P(B | A) = 1/3. Since 1/3 is not 1/2, the events are dependent: learning A occurred changes the chance of B.

With replacement, the first ball goes back before the second draw. The chance of red on the second draw remains 1/2, including after a red first draw, so P(B | A) = P(B) = 1/2. In this model, the events are independent. This illustrates why the experiment’s rules matter. [S1]

Use the product check instead. Multiply the individual probabilities and compare that result with the probability that both events happen: P(A and B) should equal P(A) × P(B) for independence. [S1]

For the same bag example with replacement, P(A and B) = (1/2)(1/2) = 1/4, which matches P(A)P(B). Without replacement, P(A and B) = (2/4)(1/3) = 1/6, while P(A)P(B) = (1/2)(1/2) = 1/4. The mismatch shows dependence. Keep the event definitions and experiment rules consistent when comparing the numbers.

No. Timing or wording alone does not prove independence. Ask whether the information that one event occurred changes the probability of the other within the model. For repeated draws, for instance, replacement matters because the first result can change what remains for the next draw. [S1]

A useful write-up is: “Let A mean [event A] and B mean [event B]. Under the assumption that [relevant experiment rule], P(A) = [value] and P(A | B) = [value]. Because [they are equal / they differ], the events are [independent / dependent] in this model.” State assumptions rather than filling in missing rules. The reference explains the independence tests and the draw examples; it does not establish the probabilities for a different experiment, so calculate those from that problem’s stated setup.