Slides calculate probability of a given outcome

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 Parti

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Calculate Probability of a Given

OutcomePrinciples of Cost Analysis and

Management

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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

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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

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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

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• 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%

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• 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%

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3 One Head and one Tail

4 One Tail and one Head

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Defining Outcomes

• What is the probability of each outcome?

Outcome Possible Ways to Achieve

Outcome /Total = Probability%

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Check on Learning

• What is the probability of an impossible

event?

• The sum of the probabilities of all possible

outcomes must be equal to?

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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?

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The Dice Game

Outcome Possible Ways to Achieve Number of Ways Probability

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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

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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?

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Practice Problems

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?

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Practice Problems

probability of a total divisible by 4? By 3?

• How would you approach this problem?

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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?

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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

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

• Positive outcome = at least one Head

• Negative outcome = no Heads

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• 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%

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Practice Problems

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?

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Practice Problems

• The probability of passing a certain class is

known to be 80% What is the probability of NOT passing?

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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?

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Independent Scenarios

• The probability that two independent outcomes

will BOTH occur is equal to the product of both

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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%

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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?

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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%

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Probability of Bob

= 60% * 95%

95%

= 60%

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Probability of Bob

= 60% * 95%

95%

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Probability of Bob

= 60% * 95%

95%

= 60%

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Probability of Bob

= 60% * 95%

95%

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Probability of Bob

= 60% * 95%

95%

= 60%

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• 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

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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

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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?

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Practical Exercise

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