PB2 Multistage Events and Applications of Probability Terminology: Arrangement multistage event trial expected number probability tree diagram two-stage event experimental result sample space replacement financial expectation simulation financial gain financial loss theoretical probability tree diagram PB2 Multistage Events and Applications of Probability

Sample Space The sample space of an experiment is the set of all possible outcomes of the experiment. In probability, an event is either one outcome or a collection of outcomes. For example, when a die is rolled, the result ‘a 5’ is one of six equally likely outcomes. Obtaining the outcome ‘5’ is an event. The result ‘getting a number less than 4’ is the collection of outcomes ‘1, 2, 3’. It is also an event. A multistage event is one that is made up of simpler events. To determine the sample space for a multistage event it is often convenient to use a tree diagram.

Tree diagrams List the sample space when a coin is tossed three times.

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Fundamental Counting Theorem The total number of outcomes for a multistage event can be determined from the product of the number of choices at each stage. e.g. A brand of women’s jeans is available in 10 waist sizes, three leg lengths and with side or front pockets. How many different combinations of these features are possible?

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Ordered arrangements of n items in a line a Find the number of ways in which Erin, Lea and Mark can stand in a queue. b Check the result by listing the outcomes.

Probability If all the possible outcomes are equally likely, then the theoretical probability of an event E happening is: A coin is tossed and a die is rolled. What is the probability of getting a head and an even number?

Expected Frequency The probability of getting a head on a single throw of a coin is 1/2 . Hence, if a coin is tossed 100 times we would expect half of the tosses (50) to result in a head. When a die is rolled the probability of getting a 4 is 1/6 . Hence, if a die is rolled 600 times we would expect 1/6 of the tosses (100) to result in a 4. These are simple examples of what is called the expected frequency of an event. It is the expected number of times the event would occur. This idea is formalised as follows. The probability that a basketball player scores from a free throw is 0.75. If the player has 60 free throws, what is the expected frequency of scoring? (How many times would you expect the player to score?)

Expected Value The expected value of an event is calculated by multiplying each outcome by its probability and adding all the results together. This may be written as: e.g. Calculate the expected value when a normal die is rolled.

The table shows each outcome and its probability when 4 coins are tossed simultaneously. Calculate the expected number of heads.

Financial Expectation Financial expectation is a particular application of expected value . It is found by multiplying each financial outcome by its probability and adding the results together. The financial expectation of an event is the theoretical expected return for the event. e.g. Calculate the financial expectation if there is a 30% chance of gaining $5000 and a 10% chance of losing $3000.