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Independent and Dependent Events

Two events are independent when the result of the second event is not dependent on the results of the first event.

Example: I have 3 red marbles, 2 blue marbles and 4 white marbles in a bag:

Find the probability of picking a red marble, putting it back, and then picking a blue marble.  Since I put the marble back, I still have the same number of possible outcomes for the second event.  This means that the events are independent.  I would find the Probability of two independent events by multiplying the probabilities:

P(A and B) = P(A)  X  P(B)

P(Red then Blue with replacing the red) = P(Red)  X  P(Blue)

P(Red then Blue) =      X    =

You can reduce this probability to

Two events are dependent when the result of the second event is dependent on the results of the first event.  If I did not replace the red marble in the bag, I change the probability for choosing the next blue marble.

Find the probability of picking a red marble and then picking a blue marble without replacing the red marble first.  Since removed a marble, there are only eight marbles left in the bag.

P(A and B) = P(A)  X  P(B)

P(Red then Blue) = P(Red)  X  P(Blue)

P(red then blue w/o replacement) = X = You can reduce this probability to

Notice how the independent probability and the dependent probability differ.