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If two events are independent, the probability of one or the other (or both) happening is given by the formula

probability of event 1 + probability of event 2 - the probability that both happen

What is the probability of getting a king OR a spade in cards?

[p(king) = 4/52] + [p(spade) = 1/4] - [4/52 x 1/4] = 4/13

Tree diagrams

-Most of the time there is no need to draw a tree diagram

-When you use tree diagrams, you multiply as you go along the branches

If there is a question asking for the probability of getting at least one head there is no need to work out all the different results that have "at least one head", simply work out the probability of getting no heads at all and subtract this from 1!

Conditional probability

Hot 2 4 6
Cold 3 1 4
5 5 10

So far we have only looked at situations when the events do not affect each other, however in the table above you can see that, for example, if it is dry, it is twice as likely to be hot than if it is wet so certain outcomes affect others.

What is the probability that it hot, given that it is dry? (Answer: It is hot on 4 out of 5 dry days so 4/5)

Tossing coins a number of times etc. is NOT an example of conditional probability. The chance of tossing a head on the next turn is totally unaffected by previous results. Even if you got six heads in a row when tossing an unbiased coin, it does not make the next result any more likely to be a tail!


Examples of conditional probability

Given that the first card I drew was a king of spades, what is the probability that the next will be a king IF I DO NOT REPLACE THE FIRST CARD.

P(king of spades)  [which is 1 because I know it has happened] x P(king) [which is 3/51 because I am one king down] = 3/51

However, if the question had asked for the probability of a king of spades followed by any king, the answer is different because it is not definite that I will get the king of spades.

P(king of spades)  [which is 1/52] x P(king) [which is 3/51 because I am one king down] = 1/884


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