Probability

1. Probability Objectives When you have competed it you should * know what a ‘sample space’ is * know the difference between an ‘outcome’ and an ‘event’. * know about different ways of estimating probabilities. Key terms: Sample space, Event, Complement of event, Trial/Experiment, Outcome.

2. Probability Probability is a measure of the likelihood that something happening. Estimating Probability There are three different ways of estimating probabilities. Method A: Theoretical estimation: Use symmetry i.e. counts equally likely outcomes. e.g. The probability of head ( P(H) ) when a coin is tossed.

3. Estimating Probability Method B: Experimental estimation: Collect data from an experiment or survey. e.g. What is the estimated probability of a drawing pin landing point upwards when dropped onto a hard surface. Method C: Make a subjective estimate When we cannot estimate a probability using experimental methods or equally likely outcomes, we may need to use a subjective method. e.g. What is the estimated probability of my plane crashing as it lands at a certain airport?

4. Sample space The list of all the possible outcomes is called the sample space s of the experiment It is important in probability to distinguish experiments from the outcomes which they may generate. ExperimentPossible outcomes (H, T) Tossing a coin (1, 2, 3, 4, 5, 6) Throwing a die Guessing the answer to a four multiple choice question (A, B, C, D)

5. s A A An event An Event is a defined situation. e.g. Scoring a six on the throw of an ordinary six-sided die. 1 2 3 4 5 6 The complement of an event The event ‘not A’ is called the complement of the event. The symbol A1is used to denote the complement of A. P(A) + P(A1) = 1 A1

6. Probability of an event Example A coin is tossed twice and we are interested in the event (A) that give the same result. Solution Sample space = HH, HT, TH, TT Event A = (HH, TT) P(A) = 2/4 = ½ Note: 0  P(A)  1

7. Example 1 The possibility space consists of the integers from 1 to 25 inclusive. A is the event ‘the number is a multiple of 5’. is the event ‘the number is a multiple of 3’. An integer is picked at random. Find (a) P(A), (b) P(B1)

8. Solution (a) Possibility space n(s) = 25 Number of outcomes in event A n(A) = 5 (5, 10, 15, 20 and 25) = 5/25 = 1/5 (b) P(B1) = 1 – P(B) = 1 - 8/25 = 17/25

9. Example 2 • A cubicle die, number 1 to 6, is weighted so that a six is • three times as likely to occur as any other number. • Find the probability of • a six accurring, • an even number occurring.

10. Solution Possibility space n(s) = {1, 2, 3, 4, 5, 6, 6, 6, } = n(s) = 8 (a) P( a six) = 3/8 (b) P( an even number) = 5/8

11. E, 420 A, 240 B, 990 D, 1200 C Example: A car manufacturer carried out a survey in which people were asked which factor from the following list influenced them when buying a car: A: Colour B: Service cost C: Safety D: Fuel economy E: Extras Find the probability that someone who said safety will win the prize. The names of those who took part were then placed in a prize draw.

12. E, 420 A, 240 B, 990 D, 1200 C Solution: C = 360 – ( 24 + 99 + 120 + 42 ) = 75 P(C) = 75/360 = 25/120 = 5/24