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There are multiple ways of finding probabilities of combined events. Sample space, two way tables and tree diagrams are all useful in their own way but what happens when you have 3 events, 4 events, n events?

When this happens, we have to use the AND/OR statements.

AND means we multiply

The trick with this is to rewrite the question with the words AND and OR.

Remember that these are conditional probabilities so the probabilities will change.

Example : A bag contains 7 black and 3 white balls. Three balls are removed without replacement. Find the probability that...

1) I get three black balls.

This can only happen one way...

black AND a black AND a black.

If we write this using fractions we get

 7 10
x
 6 9
x
 5 8
=
 210 630

This fraction will of course, cancel down to 1/3

2) I get 2 black and 1 white balls.

There are actually three different ways this can happen

black AND a black AND a white.

OR

black AND a white AND a black.

OR

A white AND a black AND a black.

The trick here is to know that each one of these probabilities will actually be the same (try it, you'll see it's true) so we only have to find one.

Lets find black AND a black AND a white.

 7 10
x
 6 9
x
 3 8
=
 126 630

We can now say that the probability of getting two blacks and a white is...

 126 630
+
 126 630
+
 126 630
=
 378 630

This will cancel down nicely to 3/5

3) The probability of getting at least one white.

For this, you could work out the probability of each one that satisfies this and add them all together (There's 7 ways this can happen)

or

You can use the fact that P(at least 1) = 1 - P(none)

So P(at least 1 white) = 1 - P(no whites)

P(at least 1 white) = 1 - P(3 blacks)

We worked out earlier that P(3 blacks) = 1/3

So we can say that the probability of getting at least one white = 1 - 1/3 = 2/3

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