Dependent vs Independent Events

In our day-to-day life, we come across events with uncertainty. For example, a chance of winning a lottery that you buy or a chance of getting the job that you applied. Fundamental theory of probability is used to determine mathematically the chance of happening something. Probability is always associated with random experiments. An experiment with several possible outcomes is said to be a random experiment, if the outcome on any single trial cannot be predicted in advance. Dependent and independent events are terms used in probability theory.

An event B is said to be independent of an event A, if the probability that B occurs is not influenced by whether A has occurred or not. Simply, two events are independent if the outcome of one does not affect the probability of occurrence of the other event.  In other words, B is independent of A, if P(B) = P(B|A). Similarly, A is independent of B, if P(A) = P(A|B). Here, P(A|B) denotes the conditional probability A, assuming that B has happened.  If we consider rolling of two dice, a number showing up in one die has no effect on what has come up in the other die.

For any two events A and B in a sample space S; the conditional probability of A, given that B has occurred is P(A|B) = P(A∩B)/P(B). So that, if event A is independent of event B, then P(A) = P(A|B) implies that P(A∩B) = P(A) x P(B). Similarly, if P(B) = P(B|A), then P(A∩B) = P(A) x P(B) holds. Hence, we can conclude that the two events A and B are independent, if and only if, condition P(A∩B) = P(A) x P(B) holds.

Let us assume that we roll a die and toss a coin simultaneously. Then the set of all possible outcomes or the sample space is S={(1, H) ,(2, H) ,(3 , H) ,(4, H) ,(5, H) ,(6, H) ,(1, T) ,(2, T) ,(3, T) ,(4, T) ,(5, T) ,(6, T) }. Let event A be the event of getting heads, then the probability of event A,  P(A) is 6/12 or 1/2, and let B be the event of getting a multiple of three on the die. Then P(B)= 4/12 = 1/3. Any of these two events has no effect on the occurrence of the other event. Hence, these two events are independent. Since the set (A∩B) = {(3,H), (6,H)}, the probability of an event getting heads and multiple of three on die, that is P(A∩B) is 2/12 or 1/6. The multiplication, P (A) x P(B) is also equals to 1/6.  Since, the two events A and B holds the condition, we can say that A and B are independent events.

If the outcome of an event is influenced by the outcome of the other event, then the event is said to be dependent.

Assume that we have a bag that contains 3 red balls, 2 white balls, and 2 green balls. The probability of drawing a white ball randomly is 2/7. What is the probability of drawing a green ball? Is it 2/7?

If we had drawn the second ball after replacing the first ball, this probability will be 2/7. However, if we do not replace the first ball that we have taken out, then we have only six balls in the bag, so the probability of drawing a green ball is now 2/6 or 1/3. Therefore, the second event is dependent, since the first event has an effect on the second event.

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