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.

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. Probability

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?

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. Experiment Possible outcomes Tossing a coin (H, T) Throwing a die (1, 2, 3, 4, 5, 6) Guessing the answer to a four multiple choice question (A, B, C, D)

The complement of an event The event ‘not A’ is called the complement of the event. The symbol A 1 is used to denote the complement of A. P(A) + P(A 1 ) = 1 An Event is a defined situation. e.g. Scoring a six on the throw of an ordinary six- sided die. An event A s A A1A1

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

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(B 1 )

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

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) a six accurring, (b) an even number occurring.

Solution Possibility space n(s) = {1, 2, 3, 4, 5, 6, 6, 6, } = n(s) = 8 (a) P( a six) = 3/83/8 (b) P( an even number) = 5/85/8

A, 24 0 B, 99 0 C D, E, 42 0 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 The names of those who took part were then placed in a prize draw. Find the probability that someone who said safety will win the prize.

A, 24 0 B, 99 0 C D, E, 42 0 Solution: C = 360 – ( ) = 75 P(C) = 75 / 360 = 25 / 120 = 5 / 24