How do we find the probability of an event?

If two events are mutually exclusive (independent), then the outcome of one event have no effect on the outcome of the second. For example, if I flip a coin twice, whether the first toss landed heads of tails has no impact on how the second toss landed. The probability of getting heads first time was ½, and the probability of getting heads the second time was still ½.

However, it is possible for two events to not be mutually exclusive (dependent). For example, if I have 10 sweets in a pack, 5 red and 5 blue, then the probably of my selecting a red sweet first time at random is 5/10 (or ½). If I eat the sweet, then there are only 9 left in the pack, and only 4 of them are red, so if I select a second sweet at random, the probably of it being red is now 4/9. The probably of my selecting a red sweet second time has changed, depending on what I selected the first time.

These types of question are best solved using a probability tree diagram.


A bag contains 4 red balls, 6 black balls. A ball is taken out of the bag at random and not replaced. Then another ball is taken out.

This question is similar to the independent case, however since the ball is now not being replaced, the probability of taking a second ball depends on what ball was taken out first.

If first a black ball is taken out then a red ball is taken out, the probability of taking a black ball first is still 6/10. However now there are only 5 black balls and 4 red balls in the bag. So the probability of now taking a red ball is 4/9. So the probability of taking a black ball then a red ball is 6/10 x 4/9 = 24/90 = 4/15

To find the probability of getting a ball with each colour i.e. what is the probability of getting first a black then a red ball, and the probability of getting a red then a black ball. From our diagram we can see the probability of both is 4/15. So we need to add these to get 8/15. Remember all the probabilities added together on each branch has to add up to 1 (if not ...something has gone wrong!)


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AQA Unit 1 November 2011 (H) - Page 10, Question 8 

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