Let us consider two events, A and B. If the probability that event A occurs has no effect on the probability that event B occurs, then A and B are independent events. A classic example of independent events is two tosses of the same fair coin. If a coin lands heads once, this has no influence on whether it will land heads again. The probability of landing heads on any given toss of the fair coin is ½.
It is a common error to presume that once a coin has landed heads for a number of times, this increases its probability of landing tails the next time it is tossed. If each toss is an independent event, this cannot be the case. Even if the coin has landed heads for 1000 consecutive times previously, its probability of landing heads the next time it is tossed is ½.
With two dependent events, on the other hand, the outcome of the first event affects the probability of the second. A classic example of such events would be drawing cards from a deck without replacement. A standard 52-card deck contains 4 aces. On the first draw, the probability of choosing an ace is 4/52 or 1/13. However, the probability of choosing an ace on the second draw will depend on whether an ace was selected on the first draw.
If an ace was selected on the first draw, there are 51 cards left to choose from, 3 of which are aces. So the probability of selecting an ace on the second draw is 3/51. But if an ace was not selected on the first draw, there are 4 aces left among 51 cards, so the probability of selecting an ace on the second draw is 4/51. Clearly, then, multiple drawings of cards from a deck without replacement are dependent events.
With any number of independent events, it is possible to use the multiplication rule to know the probability of some number of these events occurring. For example, if A, B, and C are independent events, and P(A) — the probability of A — is 1/3, P(B) is 3/5, and P(C) is 4/11, then the probability that A and B will occur is P(A)*P(B) = (1/3)(3/5) = 1/5. The probability that A, B, and C will occur is P(A)*P(B)*P(C) = (1/3)(3/5)(4/11) = 4/55.
It is important to only use the multiplication rule for independent events. With dependent events, the computation of probabilities for multiple events is not so straightforward and depends on the specific situation and dependence relationship among events. But further explorations of the world of probability theory will acquaint one with methods of analyzing probabilities of multiple dependent events as well.