6. Multistage events and application of probability Cambridge University Press  G K Powers 2013 Study guide Chapter 6

Multistage events Outcomes for each event are listed down the page with the events extending across the page. Each event is a new branch of the tree. The sample space is listed on the right-hand side HSC Hint – Always draw large clear tree diagrams and list the sample space on the right-hand side. Cambridge University Press  G K Powers 2013

Definition of probability Probability is the chance of something happening. The probability of the event is calculated by dividing the number of favourable outcomes by the total number of outcomes. HSC Hint – Probabilities expressed as a fraction need to be simplified if possible. Cambridge University Press  G K Powers 2013

Fundamental counting principle The fundamental counting principle is used to determine the total number of outcomes for a multistage event. It states if we have ‘p’ outcomes for first event and ‘q’ outcomes for the second event, then the total number of outcomes for both events is p × q. Number of outcomes (two events) = p × q. p – Number of outcomes of the first event. q – Number of outcomes of the second event. HSC Hint – Multiply the number of outcomes for each event to determine the number of items in the sample space. Cambridge University Press  G K Powers 2013

Ordered selections Ordered selections or a permutation occurs when a selection is made from a group of items and the order is important. AB is different to BA. Permutation nPr n items available for selection and r items to be selected HSC Hint – Read the question and check whether the selection is ordered. Cambridge University Press  G K Powers 2013

Unordered selections Unordered selections or a combination occurs when a selection is made from a group of items and the order is not important. AB is the same as BA. Combination nCr n items available for selection and r items to be selected HSC Hint – Read the question and check whether the selection is unordered. Cambridge University Press  G K Powers 2013

Probability trees: Product rule The probability of two events occurring is equal to the product of the probability of each event. P(AB) = P(A) × P(B) P(AB) ‒ Probability of event A and event B. P(A) ‒ Probability of event A. P(B) ‒ Probability of event B. HSC Hint – Probability questions involving two events and the word ‘and’ often require the product rule. Cambridge University Press  G K Powers 2013

Probability trees: Addition rule The probability of one event or a second event is equal to the sum of the probabilities of each event. P(A or B) = P(A) + P(B) P(A or B) ‒ Probability of event A or event B. P(A) ‒ Probability of event A. P(B) ‒ Probability of event B. HSC Hint –Probability questions involving two events and the word ‘or’ often require the addition rule. Cambridge University Press  G K Powers 2013

Expected outcomes Expected outcome is the number of times the outcome should occur. It may not be a whole number. The expected outcome is an estimate of what to expect. Expected outcome = P(E) × Number of trials P(E) – Probability of the event. HSC Hint – The expected outcome is not always a whole number. Remember it is an approximation to the number of outcomes. Cambridge University Press  G K Powers 2013

Expected value Expected value = Sum all results [P(E) × outcome] Financial expectation = Sum all results [P(E) × Financial outcome] P(E) – Probability of the event or financial outcome Financial outcome ‒ Positive if winning and negative if losing. HSC Hint – Financial outcomes that win money are positive. Financial outcomes that lose money are negative. Cambridge University Press  G K Powers 2013