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Revision:ProbabilityTSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Probability
IntroductionProbability is the likelihood or chance of an event occurring.
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\displaystyle \mathsf{Probability = \frac{the\ number\ of\ ways\ of\ achieving\ success\}{the\ total\ number\ of\ possible\ outcomes\}
We write P(heads) = ½ . The probability of something which is certain to happen is 1. The probability of something which is impossible to happen is 0. The probability of something not happening is 1 minus the probability that it will happen.
Single EventsExampleThere are 6 beads in a bag, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow? The probability is the number of yellows in the bag divided by the total number of balls, i.e. 2/6 1/3.
ExampleThere is a bag full of coloured balls, red, blue, green and orange. Balls are picked out and replaced. John did this 1000 times and obtained the following results: Number of blue balls picked out: 300 Number of red balls: 200 Number of green balls: 450 Number of orange balls: 50 a) What is the probability of picking a green ball? b) If there are 100 balls in the bag, how many of them are likely to be green? a) For every 1000 balls picked out, 450 are green. Therefore P(green) 450/1000 0.45 b) The experiment suggests that 450 out of 1000 balls are green. Therefore, out of 100 balls, 45 are green (using ratios).
Multiple EventsWhen working out what the probability of two things happening is, a probability/ possibility space can be drawn. For example, if you throw two dice, what is the probability that you will get: a) 8, b) 9, c) either 8 or 9?
The probability space shows us that when throwing 2 dice, there are 36 different possibilities (36 squares). With 5 of these possibilities, you will get 8. Therefore P(8) 5/36 . b) The red blobs indicate the ways of getting 9. There are four ways, therefore P(9) 4/36 1/9. c) You will get an 8 or 9 in any of the 'blobbed' squares. There are 9 altogether, so P(8 or 9) 9/36 1/4 . Another way of representing 2 or more events is on a probability tree.
ExampleThere are 3 balls in a bag: red, yellow and blue. One ball is picked out, and not replaced, and then another ball is picked out.
The probability is 1/3 for each of these. If a red ball is picked out, there will be two balls left, a yellow and blue. The probability the second ball will be yellow is 1/2 and the probability the second ball will be blue is 1/2. The same logic can be applied to the cases of when a yellow or blue ball is picked out first. In this example, the question states that the ball is not replaced. If it was, the probability of picking a red ball (etc.) the second time will be the same as the first (i.e. 1/3).
The AND and OR rulesIn the above example, the probability of picking a red first is 1/3 and a yellow second is 1/2. The probability that a red AND then a yellow will be picked is 1/3 × 1/2 1/6 (this is shown at the end of the branch). The probability of picking a red OR yellow first is 1/3 + 1/3 2/3. When the word 'and' is used we multiply. When 'or' is used, we add. On a probability tree, when moving from left to right we multiply and when moving down we add.
ExampleWhat is the probability of getting a yellow and a red in any order? This is the same as: what is the probability of getting a yellow AND a red OR a red AND a yellow. P(yellow and red) 1/3 × 1/2 1/6 P(red and yellow) 1/3 × 1/2 1/6 P(yellow and red or red and yellow) 1/6 + 1/6 1/3
CommentsProbably far too much in too brief notes for the whole topic - much more needed on probability trees and multiple events as these are very common exam questions at higher level. |
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