calculate probability of a given outcome © dale r. geiger 20111

© Dale R. Geiger 2011 1

Calculate Probability of a Given Outcome

Principles of Cost Analysis and Management


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


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


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 OutcomeOccurrence 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



• 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%



• 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%



• 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%


Defining Outcomes

• Using two different coins, what are the possible outcomes?

1. Two Heads 2. Two Tails3. One Head and one Tail4. One Tail and one Head


Defining Outcomes

• What is the probability of each outcome?Outcome Possible Ways to Achieve

Outcome/Total = 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


Check on Learning

• What is the probability of an impossible event?

• The sum of the probabilities of all possible outcomes must be equal to?


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?


The Dice GameOutcome 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%


Calculating Probability

1. Define all possible or relevant outcomes2. Determine number of ways of achieving the particular

outcome3. Determine total number of ways of achieving all possible

or relevant outcomes4. 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


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?


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?


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?


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?


Check on Learning

• What is the first step in calculating probability?

• What is the formula for calculating probability?


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



• Example: When tossing two coins, what is the probability of at least one Head?

• Positive outcome = at least one Head• Negative outcome = no Heads



• 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%


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?


Practice Problems

• The probability of passing a certain class is known to be 80%. What is the probability of NOT passing?


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?


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


Independent Scenarios

• When tossing two coins, what is the probability of achieving two Heads twice in a row? • The 2nd toss is not dependent upon the 1st

• Probability of two Heads on the 1st toss = 25%

• Probability of two Heads on the 2nd toss = 25%

• Probability of two Heads on both tosses =

25% * 25% = 6.25%


Practice Problems

• What is the probability of rolling a total of 7 twice in a row?

Probability of 7 * Probability of 716.7% * 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%


Practice Problems






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 passing95% * 60% = 57%


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?


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 BProbability of Condition B


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?


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 passingProbability of Bob passing

= Probability of Ted * Probability of Bob Probability of Bob

= 60% * 95% 95%

= 60%






= 60% * 95% 95%

= 60%



• 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 BProbability of B


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


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?


Practical Exercise

calculate probability of a given outcome © dale r. geiger 20111

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