Probability

1
Probability

• Formal study of uncertainty
• The engine that drives statistics
• Primary objective of lecture unit 4 use the
  rules of probability to calculate appropriate
  measures of uncertainty.

2
Introduction

• Nothing in life is certain
• We gauge the chances of successful outcomes in
  business, medicine, weather, and other everyday
  situations such as the lottery (recall the
  birthday problem)

3
History

• For most of human history, probability, the
  formal study of the laws of chance, has been used
  for only one thing gambling

4
History (cont.)

• Nobody knows exactly when gambling began goes
  back at least as far as ancient Egypt where
  4-sided astragali (made from animal heelbones)
  were used

5
History (cont.)

• The Roman emperor Claudius (10BC-54AD) wrote the
  first known treatise on gambling.
• The book How to Win at Gambling was lost.

6
Approaches to Probability

• Relative frequency
• event probability x/n,
• where x of occurrences of event of interest,
  ntotal of observations
• Coin, die tossing nuclear power plants?
• Limitations
• repeated observations not practical

7
Approaches to Probability (cont.)

• Subjective probability
• individual assigns prob. based on personal
  experience, anecdotal evidence, etc.
• Classical approach
• every possible outcome has equal probability
  (more later)

8
Basic Definitions

• Experiment act or process that leads to a single
  outcome that cannot be predicted with certainty
• Examples
• 1. Toss a coin
• 2. Draw 1 card from a standard deck of cards
• 3. Arrival time of flight from Atlanta to RDU

9
Basic Definitions (cont.)

• Sample space all possible outcomes of an
  experiment. Denoted by S
• Event any subset of the sample space S
• typically denoted A, B, C, etc.
• Simple event event with only 1 outcome
• Null event the empty set F
• Certain event S

10
Examples

• 1. Toss a coin once
• S H, T A H, B T simple events
• 2. Toss a die once count dots on upper face
• S 1, 2, 3, 4, 5, 6
• Aeven of dots on upper face2, 4, 6
• B3 or fewer dots on upper face1, 2, 3

11
Laws of Probability
12
Laws of Probability (cont.)

• 3. P(A ) 1 - P(A)
• For an event A, A is the complement of A A is
  everything in S that is not in A.

S
A'
A
13
Birthday Problem

• What is the smallest number of people you need
  in a group so that the probability of 2 or more
  people having the same birthday is greater than
  1/2?
• Answer 23
• No. of people 23 30 40 60
• Probability .507 .706 .891 .994

14
Example Birthday Problem

• Aat least 2 people in the group have a common
  birthday
• A no one has common birthday

15
Unions and Intersections
S
AÇB
A
B
AÈB
16
Mutually Exclusive (Disjoint) Events

• Mutually exclusive or disjoint events-no outcomes
  from S in common

A Ç B Æ
S
A
B
17
Laws of Probability (cont.)

• Addition Rule for Disjoint Events
• 4. If A and B are disjoint events, then
• P(A ? B) P(A) P(B)

18
Laws of Probability (cont.)

• General Addition Rule
• 5. For any two events A and B
• P(A ? B) P(A) P(B) P(A ? B)

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P(AÈB)P(A) P(B) - P(A Ç B)
S
AÇB
A
B
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Example toss a fair die once

• S 1, 2, 3, 4, 5, 6
• A even appears 2, 4, 6
• B 3 or fewer 1, 2, 3
• P(A È B) P(A) P(B) - P(A Ç B)
• P(2, 4, 6) P(1, 2, 3) - P(2)
• 3/6 3/6 - 1/6 5/6

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Laws of Probability Summary

• 1. 0 ? P(A) ? 1 for any event A
• 2. P(?) 0, P(S) 1
• 3. P(A) 1 P(A)
• 4. If A and B are disjoint events, then
• P(A ? B) P(A) P(B)
• 5. For any two events A and B,
• P(A ? B) P(A) P(B) P(A ? B)

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End of First Part of Section 4.1