Calculate Probability of a Given Outcome Principles of

Calculate Probability of a Given Outcome Principles of Cost Analysis and Management © Dale R. Geiger 2011 1

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 2

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 3

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 4

What is Probability? • The probability of an outcome must be a number between 0 and 1 (inclusive) • Probabilities are frequently stated as percentages • Probability of an impossible event is 0 or 0% • Probability of an absolutely certain event is 1 or 100% © Dale R. Geiger 2011 5

What is Probability? • Example: What are the possible outcomes when flipping a single coin? Heads • • -or- Tails 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 6

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 7

Defining Outcomes • Using two different coins, what are the possible outcomes? 1. 2. 3. 4. Two Heads Two Tails One Head and one Tail One Tail and one Head © Dale R. Geiger 2011 8

Defining Outcomes • What is the probability of each outcome? Outcome Possible Ways to Achieve /Total Outcome = Probability% 2 Heads 1 /4 = 25% 2 Tails 1 /4 = 25% 1 Head-1 Tail* 2 /4 = 50% Total 4 /4 = 100% *The combination may be 1 head-1 tail or 1 tail-1 head © Dale R. Geiger 2011 9

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

The Dice Game • What are the possible outcomes for the total of both dice when rolling a pair of dice? • Look at the results of the game to see what different outcomes occurred • It is possible to roll any of the following totals: • 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 • How many of each outcome actually occurred? • How many in proportion to the total number of rolls? © Dale R. Geiger 2011 11

The Dice Game Outcome Possible Ways to Achieve Number of Ways Probability 2 1 -1 1 1/36 or 2. 8% 3 1 -2, 2 -1 2 2/36 or 5. 6% 4 1 -3, 2 -2, 3 -1 3 3/36 or 8. 3% 5 1 -4, 2 -3, 3 -2, 4 -1 4 4/36 or 11. 1% 6 1 -5, 2 -4, 3 -3, 4 -2, 5 -1 5 5/36 or 13. 9% 7 1 -6, 2 -5, 3 -4, 4 -3, 5 -2, 6 -1 6 6/36 or 16. 7% 8 2 -6, 3 -5, 4 -4, 5 -3, 6 -2 5 5/36 or 13. 9% 9 3 -6, 4 -5, 5 -4, 6 -3 4 4/36 or 11. 1% 10 4 -6, 5 -5, 6 -4 3 3/36 or 8. 3% 11 5 -6, 6 -5 2 2/36 or 5. 6% 12 6 -6 1 1/36 or 2. 8% Total 36 36/36 or 100% © Dale R. Geiger 2011 12

Calculating Probability 1. Define all possible or relevant outcomes 2. Determine number of ways of achieving the particular outcome 3. Determine total number of ways of achieving all possible or relevant outcomes 4. Divide the number of ways of achieving the particular outcome by the total ways of achieving all possible or relevant outcomes 5. Probability = Number of ways of achieving the particular outcome Total number of ways of achieving all outcomes © Dale R. Geiger 2011 13

Practice Problems • When rolling a pair of dice, what is the probability of rolling a total divisible by 5? • Of all of the possible outcomes (2 -12), which ones are divisible by 5? • How many ways of achieving each of those? © Dale R. Geiger 2011 14

Practice Problems • When rolling a pair of dice, what is the probability of an even numbered total? • Of all of the possible outcomes (2 -12), which ones are even? • How many ways of achieving each of those? © Dale R. Geiger 2011 15

Practice Problems • When rolling a pair of dice, what is the probability of a total divisible by 4? By 3? • How would you approach this problem? © Dale R. Geiger 2011 16

Practice Problems • The bag of candy has 20 red candies, 10 yellow and 5 green. You reach in and take one. What is the probability of getting a green one? A red? A yellow? • What are the possible outcomes? • How many ways to achieve each outcome? © Dale R. Geiger 2011 17

Check on Learning • What is the first step in calculating probability? • What is the formula for calculating probability? © Dale R. Geiger 2011 18

Probability of Negative Outcome • It may not be relevant to define the probabilities of all possible outcomes • What may be relevant is to define two possible outcomes: • Positive – a particular outcome • Negative – all other outcomes • If the probability of one is known, the other can be calculated • Probability of Positive = P • Probability of Negative = 100% - P © Dale R. Geiger 2011 19

Probability of Negative Outcome • Example: When tossing two coins, what is the probability of at least one Head? • Positive outcome = at least one Head • Negative outcome = no Heads © Dale R. Geiger 2011 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? • One way: Tail-Tail • Probability of at least one Head is 3/4 or 75% • Probability of no Heads is 1/4 or 25% © Dale R. Geiger 2011 21

Practice Problems • When rolling a pair of dice, what is the probability of NOT rolling a total of 6? • Of NOT rolling a total of 7? • What is the probability of NOT rolling a number divisible by 5? © Dale R. Geiger 2011 22

Practice Problems • The probability of passing a certain class is known to be 80%. What is the probability of NOT passing? © Dale R. Geiger 2011 23

Check on Learning • How would you express the probability of NOT being struck by lightning? • What is the probability of NOT rolling a 2 when rolling two dice? © Dale R. Geiger 2011 24

Independent Scenarios • The probability that two independent outcomes will BOTH occur is equal to the product of both outcomes • Since both probabilities are less than 100%, the probability of BOTH will be less than the probability of either one alone • Examples: 80% * 60% = 48% ½*½=¼ © Dale R. Geiger 2011 25

Independent Scenarios • When tossing two coins, what is the probability of achieving two Heads twice in a row? • The 2 nd toss is not dependent upon the 1 st • Probability of two Heads on the 1 st toss = 25% • Probability of two Heads on the 2 nd toss = 25% • Probability of two Heads on both tosses = 25% * 25% = 6. 25% © Dale R. Geiger 2011 26

Practice Problems • What is the probability of rolling a total of 7 twice in a row? Probability of 7 * Probability of 7 16. 7% * 16. 7% = 2. 8% • What is the probability of rolling a total other than 7 twice in a row? Probability of not 7 * Probability of not 7 (1 – 16. 7% ) * (1 – 16. 7%) = 69. 4% © Dale R. Geiger 2011 27

Practice Problems • What is the probability of rolling a total of 7 twice in a row? Probability of 7 * Probability of 7 16. 7% * 16. 7% = 2. 8% • What is the probability of rolling a total other than 7 twice in a row? Probability of not 7 * Probability of not 7 (1 – 16. 7% ) * (1 – 16. 7%) = 69. 4% © Dale R. Geiger 2011 28

Practice Problems • What is the probability of rolling a total of 7 twice in a row? Probability of 7 * Probability of 7 16. 7% * 16. 7% = 2. 8% • What is the probability of rolling a total other than 7 twice in a row? Probability of not 7 * Probability of not 7 (1 – 16. 7% ) * (1 – 16. 7%) = 69. 4% © Dale R. Geiger 2011 29

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 = 60% * 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 = 60% * 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 = 60% * 95% = 60% © 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 = 60% * 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 = 60% * 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 = 60% * 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 © 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