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### Probability: Independent and Dependent Events

In probability, two events are said to be independent if they do not affect the probability of each other. For example, you toss two coins one after the other. The probability of getting heads on the first coin is 1/2 and that of getting tails on the second coin is 1/2. Even if you toss only one coin, or even three coins, the probability of getting either heads or tails on each coin does not change. It remains 1/2. These kind of events which do not affect the probability of each other are called independent events.

In order to understand it better, let us now discuss two events that are not independent. Consider a bag full of marbles of different colors. The bag has 5 blue marbles, 10 white marbles and 4 red marbles. Suppose you take two marbles from the bag in two consecutive draws. Suppose you drew a blue marble on the first draw. Then, after this first draw, you are left with 18 marbles (4 blue, 10 white and 4 red). Now the probability of getting a white marble in the second draw is equal to
P("white marble on second draw") = "Number of white marbles"/"Total number of marbles"
P("white marble on second draw") = 10/18 = 5/9
(If you did not understand this, it would be better if you read this post first)
Thus the probability of getting a white marble on the second draw is 5/9. But this is when the first draw gave you a blue marble. What if you got, instead of a blue marble, you got a white marble in the first draw? Will the probability of getting the white marble on the second draw remain the same? Let's see.

Suppose you draw a white marble on the first draw. Then, after the first draw, there are a total of 18 marbles in the bag out of which 5 are blue, 9 are white and 4 are red. Now, the probability of getting a white marble in the second draw is
P("white marble on second draw") = "Number of white marbles"/"Total number of marbles"
P("white marble on second draw") = 9/18 = 1/2
Notice that we got two different results for the probability of getting a white marble on the second draw here: 5/9 and 1/2. When the first marble drawn is blue, the probability of getting a white marble on the second draw is 5/9, and when the first marble drawn is white, the probability of getting a white marble on the second draw is 1/2. Thus, we see that the probability of getting a white marble in the second draw is dependent on what you get in the first draw. Thus, the probability of the second event is dependent on the outcome of the first event. These type of events are called dependent events and are not independent events.