Calculate Probability of a Given Outcome Principles of Cost Analysis and Management © Dale R Geiger 2011 The Dice Game • Divide the students into equal groups • Each group receives a pair of dice • Students will each roll the dice five times, keeping track of the total of each roll • There will be a prize for highest individual score and lowest individual score • There will be a prize for the group that finishes the task first © Dale R Geiger 2011 Terminal Learning Objective • Action: Calculate probability of a given outcome • Condition: You are a cost advisor technician with access to all regulations/course handouts, and awareness of Operational Environment (OE)/Contemporary Operational Environment (COE) variables and actors • Standard: With at least 80% accuracy: • Identify and enter relevant report data into macro enabled templates to solve Probability equations © Dale R Geiger 2011 What is Probability? • Probability is the likelihood or chance of a particular outcome in relation to all possible outcomes • Implies a division or ratio relationship: Occurrence of Particular Outcome Occurrence of All Outcomes • Defining all possible outcomes in real-life scenarios can be difficult, if not impossible • To help us understand the concept of probability we use simple examples with easily determined outcomes © Dale R Geiger 2011 What is Probability? • The probability of an outcome must be a number between and (inclusive) • Probabilities are frequently stated as percentages • Probability of an impossible event is or 0% • Probability of an absolutely certain event is or 100% © Dale R Geiger 2011 What is Probability? • Example: What are the possible outcomes when flipping a single coin? • • • • Heads -orTails What is the chance or probability of Heads? Heads is one of only two possible outcomes The probability is 1/2 or 50% (with a fair coin) Probability of Tails is also 50% © Dale R Geiger 2011 What is Probability? • The sum of the individual probabilities of all possible outcomes must equal 100% • Probability of all possible coin-flip outcomes: Heads 50% Tails 50% 100% © Dale R Geiger 2011 Defining Outcomes • Using two different coins, what are the possible outcomes? Two Heads Two Tails One Head and one Tail One Tail and one Head © Dale R Geiger 2011 Defining Outcomes • What is the probability of each outcome? Outcome Possible Ways to Achieve Outcome /Total = Probability% Heads /4 = 25% Tails /4 = 25% Head-1 Tail* /4 = 50% Total /4 = 100% *The combination may be head-1 tail or tail-1 head © Dale R Geiger 2011 Check on Learning • What is the probability of an impossible event? • The sum of the probabilities of all possible outcomes must be equal to? © Dale R Geiger 2011 10 Practice Problems • The probability that Bob will pass the course is 95% The probability that Ted will pass the course is 60% What is the probability of both Bob and Ted passing? Probability of Bob passing * Probability of Ted passing 95% * 60% = 57% © Dale R Geiger 2011 30 Practice Problems • The probability that Bob will pass the course is 95% The probability that Ted will pass the course is 60% What is the probability of both Bob and Ted passing? Probability of Bob passing * Probability of Ted passing 95% * 60% = 57% © Dale R Geiger 2011 31 Check on Learning • Even if the probabilities of two independent events are not known, what can be said about the probability of BOTH events occurring? © Dale R Geiger 2011 32 Conditional Scenarios • What is the probability of an outcome given a particular condition has already occurred? • The condition reduces the number of possible outcomes • Probability of Outcome A given Conditional Outcome B has already occurred = Probability of BOTH A and B Probability of Condition B © Dale R Geiger 2011 33 Conditional Probabilities • The probability that Ted will pass the course is 60% The probability that Bob will pass the course is 95% • Given that Bob has already passed the course, what is the probability of both Bob and Ted passing? © Dale R Geiger 2011 34 Conditional Probability • What is the desired “Outcome A”? Both pass • What is the “Condition B” or given? Bob passes Probability of BOTH Ted and Bob passing Probability of Bob passing = Probability of Ted * Probability of Bob Probability of Bob = 60% * 95% 95% = 60% © Dale R Geiger 2011 35 Conditional Probability • What is the desired “Outcome A”? Both pass • What is the “Condition B” or given? Bob passes Probability of BOTH Ted and Bob passing Probability of Bob passing = Probability of Ted * Probability of Bob Probability of Bob = 60% * 95% 95% = 60% © Dale R Geiger 2011 36 Conditional Probability • What is the desired “Outcome A”? Both pass • What is the “Condition B” or given? Bob passes Probability of BOTH Ted and Bob passing Probability of Bob passing = Probability of Ted * Probability of Bob Probability of Bob = 60% * 95% 95% = 60% â Dale R Geiger 2011 37 Conditional Probability What is the desired “Outcome A”? Both pass • What is the “Condition B” or given? Bob passes Probability of BOTH Ted and Bob passing Probability of Bob passing = Probability of Ted * Probability of Bob Probability of Bob = 60% * 95% 95% = 60% © Dale R Geiger 2011 38 Conditional Probability • What is the desired “Outcome A”? Both pass • What is the “Condition B” or given? Bob passes Probability of BOTH Ted and Bob passing Probability of Bob passing = Probability of Ted * Probability of Bob Probability of Bob = 60% * 95% 95% = 60% © Dale R Geiger 2011 39 Conditional Probability • What is the desired “Outcome A”? Both pass • What is the “Condition B” or given? Bob passes Probability of BOTH Ted and Bob passing Probability of Bob passing = Probability of Ted * Probability of Bob Probability of Bob = 60% * 95% 95% = 60% © Dale R Geiger 2011 40 Conditional Probability • If the probability of the Outcome A is truly independent of Condition B, then… • The probability of Outcome A given Conditional Outcome B will be equal to the probability of Outcome A alone: Probability of A * Probability of B Probability of B © Dale R Geiger 2011 41 What If? • What if Bob and Ted are brothers who are extremely competitive? Given that Bob has already passed the course, will the probability of Ted passing the course change? • We can’t say exactly how Bob’s passing the course will affect Ted, but it seems likely that it will • If the probability of A given B is different than the probability of A alone, then we say the two outcomes are dependent © Dale R Geiger 2011 42 Check on Learning • 10% of students receive an A in English and 15% receive an A in Math What is the probability of receiving an A in both classes? • If you have already received an A in English, what is the probability of receiving an A in Math? • Are there any other factors that might affect your actual outcome? © Dale R Geiger 2011 43 Practical Exercise © Dale R Geiger 2011 44 ... 20 Probability of Negative Outcome • What are the possible ways to achieve a positive outcome? • Three ways: Head-Head, Head-Tail, Tail-Head • What are the possible ways to achieve a negative outcome? ... Number of ways of achieving the particular outcome Total number of ways of achieving all outcomes â Dale R Geiger 2011 13 Practice Problems When rolling a pair of dice, what is the probability of. .. number of ways of achieving all possible or relevant outcomes Divide the number of ways of achieving the particular outcome by the total ways of achieving all possible or relevant outcomes Probability