Difference Between Dependent and Independent Events

Dependent vs Independent Events

在我们的日常生活中,我们遇到的事件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 eventBis said to beindependentof an eventA,if the probability thatBoccurs is not influenced by whetherAhas 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,Bis independent ofA,if P(B) = P(B|A).Similarly,Ais independent ofB,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 andBin a sample space S; the conditional probability ofA, given thatBhas 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.

What is the difference between Dependent Event and Independent Event? Two events are said to be independent events, if the two events have no effect on one another. Otherwise they are said to be dependent events. If two events A and B are independent, then P(A∩B) = P(A) . P(B)