Statistics 270 - Lecture 4

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Statistics 270 - Lecture 4
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• Last class measures of spread and box-plots
• Have completed Chapter 1
• Today - Chapter 2

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Probability

• There is a 75 chance of rain tomorrow
• What does this mean?

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Definitions

• Probability of an outcome is a numerical measure
  of the chance of the outcome occurring
• A experiment is any action whose outcome is
  uncertain
• Sample space, S, is the collection of possible
  outcomes of an experiment
• Event is a set of outcomes
• Event occurs when one of its outcomes occurs

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Example

• A coin is tossed 1 time
• S
• Describe event of getting 1 heads
• Event with one outcome is called

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Example

• A coin is tossed 2 times
• S
• Describe event of getting 1 heads and 1 tails
• Event with more than one outcome is called

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Review of Sets

• The union of two events, A and B, is the event
  consisting of outcomes that are in either A or B
  or both
• The Intersection of two events, A and B, is the
  event consisting of all outcomes that are in both
  A and B
• The complement of an event A, denoted A, is the
  set of all outcomes in the sample space that are
  not in A

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Visually

• Union
• Intersection
• Complement

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• Two sets, A and B, are said to be mutually
  exclusive if they have no events in common
• Visually

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Example

• Bag of balls has 5 red and 5 green balls
• 3 are drawn at random
• S

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Example (continued)

• A is the event that at least 2 green are chosen
• A
• B is the event that 3 green are chosen
• B

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Example (continued)

• A

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Probability

• Probability of an event is the long-term
  proportion of times the event would occur if the
  experiment is repeated many times

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Probability

• Probability of event, A is denoted P(A)
• Axioms
• For any event, A,
• P(S) 1
• If A1, A2, , Ak are mutually exclusive events,
• These imply that

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Discrete Uniform Distribution

• Sample space has k possible outcomes
  Se1,e2,,ek
• Each outcome is equally likely
• P(ei)
• If A is a collection of distinct outcomes from S,
  P(A)

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Example

• A coin is tossed 1 time
• S
• Probability of observing a heads or tails is

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Example

• A coin is tossed 2 times
• S
• What is the probability of getting either two
  heads or two tails?
• What is the probability of getting either one
  heads or two heads?

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Example

• Inherited characteristics are transmitted from
  one generation to the next by genes
• Genes occur in pairs and offspring receive one
  from each parent
• Experiment was conducted to verify this idea
• Pure red flower crossed with a pure white flower
  gives
• Two of these hybrids are crossed. Outcomes
• Probability of each outcome

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Note

• Sometimes, not all outcomes are equally likely
  (e.g., fixed die)
• Recall, probability of an event is long-term
  proportion of times the event occurs when the
  experiment is performed repeatedly
• NOTE Probability refers to experiments or
  processes, not individuals

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Probability Rules

• Have looked at computing probability for events
• How to compute probability for multiple events?
• Example 65 of SFU Business School Professors
  read the Wall Street Journal, 55 read the
  Vancouver Sun and 45 read both. A randomly
  selected Professor is asked what newspaper they
  read. What is the probability the Professor
  reads one of the 2 papers?

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• Addition Rules
• If two events are mutually exclusive
• Complement Rule