5.1 basic probability ideas definition: experiment – obtaining a piece of data definition: outcome...

5.1 Basic Probability Ideas

• Definition: Experiment – obtaining a piece of data

• Definition: Outcome – result of an experiment• Definition: Sample space – list of all possible

outcomes of an experiment• Definition: Event – collection of outcomes from

an experiment (a simple event is a single outcome)• Definition: Equally likely sample space – sample

space in which all outcomes are equally likely


• Definition: Probability – A number between 0 and 1 (inclusive) that indicates how likely an event is to occur

• Definition: n(A) – number of simple events in A• Definition: Theoretical probability of A – p(A)

= n(A)/n(S) = (# outcomes of A) ÷ (# outcomes in the equally likely sample space)


• Law of Large Numbers – as the number of experiments increases without bound, the proportion of a certain event approaches a theoretical probability

• The Law of Large Numbers does not say: If you throw a coin and get heads 10 times, that your probability of getting tails increases. P(heads) stays at 1/2


• Empirical Probability – relative frequency of an event based on past experience

p(A) = (# times A has occurred) ÷ (# observations)


• Properties of Probabilities:1. 0 ≤ p(A) ≤ 1

2. p(a) = 0 A is impossible, the event will never happen

3. p(a) = 1 A is a certain event, the event must happen

4. Let a1, a2, a3,…. an be all events in a sample space, then p(a1) + p(a2) + … + p(an) = 1

5.2 Rules of Probability

• Definition: Complement of an event A – The event that A does not occur denoted AC

• Properties:1. P(A) + p(AC) = 1

2. P(AC) = 1 – p(A)

3. P(A) = 1 – p(AC)

• Odds in favor of event A – n(A):n(AC) orn(A) ÷ n(AC)


• Example: p(A) = 1/3 then p(AC) = 1 – p(A) = 2/3

odds in favor of A = p(A):p(AC) = 1/3:2/3 = 1:2

odds against A = p(AC):p(A) = 2/3:1/3 = 2:1• Probabilities from odds in favor –

odds in favor = s:f (successes to failures)p(A) = s/(s + f)p(AC) = f/(s + f)


• Joint probability tables – displays possible outcomes and their likelyhood of occurrence

• Example: Given the following table of data:

Coke Pepsi

Male 13 31

Female 22 14


• Probability table for example (total of 80 people in the sample):

Coke Pepsi

Male 13/80 = .1625 31/80 = .3875

Female 22/80 = .275 14/80 = .175


• Simple probability tree:

root

Boy

Girl

branches


• Probability trees are useful when events do not have the same probability (there is no equally likely sample space)

• Problem solutions involving trees can become long if many branches are to be calculated (similar to the brute force method in section 4.5)

5.3 Probabilities of Unions and Intersections

• Definition: The union of two events A and B is the event that occurs if either A or B or both occur in a single experiment. The union of A and B is denoted A BExample: (rolling a die – getting an even number or a perfect square)

1

2 4

3

6

5


• Definition: The intersection of two events A and B is the event that occurs if both A and B occur in a single experiment. The intersection of A and B is denoted A BExample: (rolling a die – getting an even number and a perfect square)

1

2 4

3

6

5


• Definition: mutually exclusive or disjoint events – events for which A B = (where represents an event with no elements)

• If A and B are mutually exclusive, then:

1. P(A B) = 0

2. P(A B) = P(A) + P(B)

• Union Principle of Probability:

P(A B) = P(A) + P(B) - P(A B)

5.4 Conditional Probability and Independence

• Definition: The conditional probability of A given B is the probability of A occurring given that B has already occurred – denoted P(AB)When outcomes are equally likely:

P(AB) = n(AB)n(B)

• Conditional Probability Formula (outcomes not necessarily equally likely)

P(AB) =P(AB)

P(B)


• Multiplication Principal:P(A B) = P(B) P(AB)

• Tree diagrams – useful for conditional probability because each section of a branch is a probability conditional by the previous branches


• Independence: Two events A and B are said to be independent if the occurrence of A does not affect P(B) and vice versa.

A & B are independent if: P(AB) = P(A) or P(BA) = P(B)

• Multiplication Principle for Independent Events:

A & B are independent events P(A B) = P(A) P(B)

5.5 Bayes’ Formula

• Bayes formula for 2 cases:

P(AB) = P(A) P(BA)

P(A) P(BA) + P(AC) P(BAC)

5.5 Bayes’ Formula

• Bayes formula for n disjoint events:

P(AiB) = P(Ai) P(BAi)

P(A1) P(BA1) + P(A2) P(BA2) + … + P(An) P(BAn)

5.6 Permutations and Combinations

• Multiplication Principle – given a tree with the number of choices at each branch being m1, m2, m3, … mn, then the number of possible occurrences is:

m1 m2 m3 … mn


• Permutations: The number of arrangements of r items from a set of n items.

Note: Order matters.

nPr =n!

(n – r)!


• Combinations: The number of subsets of r items from a set of n items.

Note: Order does not matter.

nCr =n!

(n – r)! r!

5.6 Permutations and Combinations- summary of counting formulas

With replacement(order matters)

Without replacement

Order matters(arrangements)

Order doesnot matter(subsets)

Multiplicationprincipal

Permutation Combination

5.7 Probability and Counting Formulas

• Example: A bag contains 4 red marbles and 3 blue marbles. Find the probability of selecting:

a. Two red marbles

b. Two blue marbles

c. A red marble followed by a blue marble

P(2 red) = # ways to pick 2 red

# ways to pick any 2 marbles= 4C2

7C2

= 6/21 = 2/7


P(2 blue) = # ways to pick 2 blue

# ways to pick any 2 marbles

= 3C2

7C2

= 3/21 = 1/7

P(red then blue) = chance of picking red on first chance of picking blue on second

= 4/7 3/6 = 2/7


• Birthday Problem: Suppose there are n people in a room. Find the formula for the probability that at least two people have the same birthday.

Note: P(at least two birthdays the same) = 1 – P(no two birthdays the same)

# ways for n people to have birthdays = 365n

# ways for for n birthdays without repeats = 365Pn

answer = 1 – (365Pn 365n)

5.8 Expected Value

• Expected Value – for a given sample space with disjoint outcomes having probabilities p1, p2, p3, … pn and a value (winnings) of x1, x2, x3, … xn , then the expected value of the sample space is:

x1 p1 + x2 p2 + x3 p3 +…. xn pn

• Definition: A game is said to be fair if the cost of participating equals the expected winnings.– Expected winnings < cost unfair to participant– Expected winnings > cost unfair to organizers

5.9 Binomial Experiments

• Definition: Binomial Experiment1. The same trial is repeated n times.

2. There are only 2 possible outcomes for each trial – success or failure

3. The trials are independent so the probability of success or failure is the same for each trial.

5.9 Binomial Experiments

• Binomial Probabilities:P(x successes) = nCr px (1-p)n-x with x = 0,1,2,…,n

where n is the number of trials, p is the probability of success, and x is the number of successful trials

• Expected value of a binomial experiment = n pnote: The most likely outcome “tends to be close to” the expected value.

5.1 basic probability ideas definition: experiment – obtaining a piece of data definition: outcome...

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