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In these lessons, we will learn how to calculate probability without replacement (dependent events) and how to use a probability tree diagram.

Related Topics: More Probability Lessons

**What is Probability without Replacement or Dependent Probability?**

In some experiments, the sample space may change for the different events. For example,
a marble may be taken from a bag with 20 marbles and then a second marble is taken
**without replacing** the first marble. The sample space
for the second event is then 19 marbles instead of 20 marbles. This is called probability
without replacement or dependent probability. We can use a tree diagram to help us find the probability without replacement.

Step 1: Draw the Probability Tree Diagram and write the probability of each branch. (Remember that the objects are not replaced)

Step 2: Look for all the available paths (or branches) of a particular outcome.

Step 3: Multiply along the branches and add vertically to find the probability of the outcome.

* Example: *

A jar consists of 21 sweets. 12 are green and 9 are blue. William picked two sweets at random.

a) Draw a tree diagram to represent the experiment.

b) Find the probability that

i) both sweets are blue.

ii) one sweet is blue and one sweet is green.

c) William randomly took a third sweet. Find the probability that:

i) all three sweets are green?

ii) at least one of the sweet is blue?

* Solution: *

Check that the probabilities in the last column add up to 1.

b) i)

P(both sweets are blue) = P(B, B)

ii)

P(one sweet is blue and one sweet is green) = P(G, B) or P(B, G)

c) i) P(all three sweets are green) = P(G, G, G)

ii) P(at least 1 sweet is blue) = 1 – P(all three sweets are green)

This video explains the difference between probability with replacement (independent events) and probability without replacement (dependent events). How to use a probability tree diagram.

Adam has a bag containing four yellow gumdrops
and one red gumdrop. he will eat one of the gumdrops, and a few minutes later, he will eat a second gumdrop.

a) Draw the tree diagram for the experiment.

b) What is the probability that Adam will eat a yellow gumdrop first and a green gumdrop second?

c) What is the probability that Adam will eat two yellow gumdrops?

d) What is the probability that Adam will eat two gumdrops with the same color?

e) What is the probability that Adam will eat two gumdrops of different colors?

A jar contains 4 black marbles and 3 red marbles. Two marbles are drawn without replacement.

a) Draw the
tree diagram for the experiment.

b) Find probabilities for P(BB), P(BR), P(RB), P(WW), P(at least one Red), P(exactly one red)

Two marbles are drawn without replacement from a jar containing 4 black and 6 white marbles.

a) Draw the tree diagram for the experiment.

b) Find the probabilities for P(at least one black marble), P(same color), P(BW), P(exactly one black marble)

A bag contains 5 blue balls and 4 red balls. A ball is picked and not replaced. What is the probability of picking at least one red ball?

A visual tutorial on how to calculate probability with and without replacement using marbles.

The following video shows how to calculate probability without replacement or dependent probability.

Andrea has 8 blue socks and 4 red socks in her drawer. She chooses one sock at random and puts it on. She then chooses another sock without looking. Find the probability of the following event P(red, then red).

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.