# Probability Without Replacement

## What is Probability Without Replacement?

As then name says, it is a probability where something is not replaced. For example, if we pick 2 marbles from a bag there are
different possibilities of what we could do:
•
Probability With Replacement
We take a marble put it back into the bag and pick another one.
•
Probability Without Replacement
We take a marble. Then we take a second a second marble (without having
replaced the first one). The probabilities of the second pick will be different as there is one less marble in the bag.
To explain how to deal with probability without replacement. We will have a look at example three from the probability lesson, this time
not replacing the marbles in the bag.
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Example 3
We now have a bag with 12 marbles (2 red, 4 blue, 6 green). We have to pick twice (not replacing the 1st marble) Find:
a) P (Same Colour Twice)
b) P (Not Blue)
a) To find the answer to part a we have to look at all the possibilities where we get the same colour twice: RED & RED, BLUE & BLUE
and GREEN & GREEN. We then have to calculate the probabilities for these combined events (working out in the red boxes). Lastly, we
have to add these probabilities.
The solution is P (Same Colour Twice) = 1/3
b) We need to find all the possibilities that do not contain blue. These are:
•
RR
•
RG
•
GR
•
GG
We need to find the probability of all these combined events. The working out is in the blue boxes. Now we simply need to add all these
combined probabilities together.
The solution is P (Not Blue) = 14/33

Pick 1 (1)

Pick 2 (2)

### Explanation

Hopefully you have understood the difference but here are just the main points you need to consider:
•
After the first pick there are less marbles in the bag since the marble has not been replaced. This is why x/12 changes to x/11.
•
If colour x has been picked during pick 1, there is one less marble of this colour in the bag.
•
If there were more than two picks, this pattern would continue.

## Explore Ultimate Maths

You should now be able to do simple and complex probability problems. We recommend that you have a look at our presenting data
lessons. If you would like to chose a completely different topic, feel free to visit our library. Please share this page if you like it or found it
helpful!

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Chapter 21.3: Learning Outcomes
Students will learn what the difference between probability with and
without replacement is!
Students will learn how to calculate probability without replacement!

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