Hi there. This math post for young students is on basic probability without replacement. Probability with replacement is usually associated with picking marbles randomly from a bag. We also work with multiple events where probabilities are multiplied. Students should be familiar with multiplying fractions.

### Topics

- Probability Without Replacement
- Practice Problems
- Answers To Practice Problems

### Probability Without Replacement

With probabilities with replacement, we will deal with randomly selecting marbles from a bag. Any marbles that are selected are not put back in the bag. This is sampling without replacement. In this section, keep in mind the total number of marbles left in the bag and the number of marbles left for each colour.

**Example One - Picking Two Marbles From A Bag**

In a bag of marbles there are 3 green marbles, 2 red marbles, and 5 blue marbles. You randomly select two marbles from the bag where the first marble is not put back into the bag. What is the probability of getting two blue marbles?

The probability of selecting a blue marble in the first pick is 5 out of 10 which is 50 percent or a half. This is because there were 5 blue marbles available and the total number of marbles in the bag is 10.

After the first marble pick, the blue marble is not put back in the bag. This means that the number of marbles in the bag that can be selected is nine. In addition, the number of blue marbles remaining is four. The probability of getting a second blue marble is 4 out of 9.

Multiplying the two fractions of 5/10 and 4/9 gives (5 x 4) / (10 x 9) = 20 / 90 = 2/9.

**Example Two - Picking Three Marbles From A Bag**

We have the same bag of marbles from the first example where there are 3 green marbles, 2 red marbles, and 5 blue marbles. You randomly select three marbles from the bag where the first marble and second marble is not put back in the bag. What is the probability of selecting a red marble, a green marble and a blue marble in the same order?

Obtaining a red marble in the first pick has a probability of 2 out of 10 or 1/5.

In the second pick, the number of red marbles is now one and the total number of marbles is 9. Selecting a green marble has a probability of 3/9 or 1/3.

In the third and last pick, the number of green marbles is 2, the number of red marbles is 2 and the number of blue marbles is still 5. The total number of marbles left in the bag is now 8. The probability of selecting a blue marble in the third pick is 5/8.

Multiplying the probabilities together gives 1/5 x 1/3 x 5/8 which is 5 / 120 = 1 / 24.

### Practice Problems

You have 8 marbles in a bag. There are 2 red marbles, 3 blue marbles and 3 yellow marbles.

What is the probability of getting a red and blue marble in the first two picks in this order? The first marble is not put back into the bag.

What is the probability of getting a blue marble and a red marble in the first two picks in this order? The first marble is not put back into the bag. Compare this answer to the one in question one.

In two tries, what is the probability of selecting all two red marbles? Assume sampling without replacement where the marbles are not put back in the bag.

In three tries, what is the probability of selecting all three yellow marbles? Assume sampling without replacement as usual.

### Answers To Practice Problems