Explain the Dependent Events?

Events are called dependent events when the outcome of one event influences the outcome of the second event.

P(A and B) = P(A) P(B following A)

For example, out of a bag that has 2 red marbles, 3 green marbles, and 4 blue marbles, what is the probability of picking a red marble, then picking a green marble, without replacing the red marble?

P(red) = 2/9
P(picking green, without putting red back) = 3/8
P(red and green) = 2/9 x3/8 = 6/72 = 1/12(about 8% chance)
Notice that the total possible outcomes for the second event became 8, because after picking a red marble, there was one less than 9 marbles to choose from.

Examples:

Out of a deck of playing cards, what's the probability of choosing:
a. an ace, then another ace without replacement of the first ace?
b. a heart, then a spade, without replacement of the heart?
c. a face card, then a seven, without replacement of the face card?

Solution
a. P(ace) = 4/52
P(second ace) = 3/51
P(ace, another ace) = 4/52 x3/51 = 12/2652 = 1/221
b. P(heart)= 13/52 = 1/4
P(spade)= 13/51
P(heart, spade)= 1/4x13/51 = 13/204
c. P(face card)= 12/52
P(seven)= 4/51
P(face card, seven)= 12/52x4/51 = 3/13 x4/51 = 12/663 = 4/221