What is the concept of independence in probability?

Independence in probability means that the occurrence of one event does not affect the probability of another event.

When two events A and B are independent, the probability of both events occurring is the product of their individual probabilities: P(A and B) = P(A) x P(B). This is known as the multiplication rule of probability.

For example, if we toss a fair coin twice, the probability of getting heads on the first toss is 1/2 and the probability of getting heads on the second toss is also 1/2. Since the two events are independent, the probability of getting heads on both tosses is: P(heads on first toss and heads on second toss) = P(heads on first toss) x P(heads on second toss) = 1/2 x 1/2 = 1/4.

On the other hand, if two events A and B are dependent, the probability of both events occurring is given by the conditional probability: P(A and B) = P(A|B) x P(B), where P(A|B) is the probability of A given that B has occurred. In this case, the multiplication rule does not apply.

For example, if we draw two cards from a standard deck of 52 cards without replacement, the probability of getting a king on the first draw is 4/52 and the probability of getting a king on the second draw given that the first card was not a king is 3/51. Therefore, the probability of getting two kings is: P(king on first draw and king on second draw) = P(king on first draw) x P(king on second draw given that the first card was not a king) = 4/52 x 3/51 = 1/221.