DEPENDENT EVENTS

Events are dependent if the outcome of one event affects the outcome of another. For example, if you draw two colored balls from a bag and the first ball is not replaced before you draw the second ball then the outcome of the second draw will be affected by the outcome of the first draw.

If A and B are dependent events, then the probability of A happening AND the probability of B happening, given A, is P(A) × P(B after A).

P(A and B) = P(A) × P(B after A)

P(B after A) can also be written as P(B | A)

then P(A and B) = P(A) × P(B | A)

Example:

A purse contains four $5 bills, five $10 bills and three $20 bills. Two bills are selected without the first selection being replaced. Find P($5, then $5)

Solution:

There are four $5 bills.

There are a total of twelve bills.

P($5) =

The result of the first draw affected the probability of the second draw.

There are three $5 bills left.

There are a total of eleven bills left.

P($5 after $5) =

P($5, then $5) = P($5) · P($5 after $5) =

The probability of drawing a $5 bill and then a $5 bill is

Example:

A bag contains 6 red, 5 blue and 4 yellow marbles. Two marbles are drawn, but the first marble drawn is not replaced.

a) Find P(red, then blue).

b) Find P(blue, then blue)

Solution:

a) There are 6 red marbles.

There are a total of 15 marbles.

P(red) =

The result of the first draw affected the probability of the second draw.

There are 5 blue marbles.

There are a total of 14 marbles left.

P(blue after red) =

P(red, then blue) = P(red) · P(blue after red) =

The probability of drawing a red marble and then a blue marble is

b) There are 5 blue marbles.

There are a total of 15 marbles.

P(blue) =

The result of the first draw affected the probability of the second draw.

There are 4 blue marbles left.

There are a total of 14 marbles left.

P(blue after blue) =

P(blue, then blue) = P(blue) · P(blue after blue) =

The probability of drawing a red marble and then a blue marble is

The following videos show more examples of calculating the probability of dependent events.

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