STAT 211 Topic 2

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Lecture 3

Lecture Notes

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Probability

Sample Space

All possible outcomes of an experiment:

EX flipping a coin:
EX rolling a die:

Event

Any subset of outcomes contained in sample space

EX flipping a coin: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=\{H\}, \{T\}, \{H,T\}, ...}

Set Theory

For two events Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}

Union (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cup B} )
all outcomes that are in A, B, or both
Intersection(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B} )
all outcomes that are in A and B
Complement (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A'} )
all outcomes that are not in A
Mutually Exclusive
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B=\{\}}

Properties

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le P(A) \le 1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P({}) = 0, P(S) = 1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A') = 1-P(A)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)}
  • If A and B are mutually exclusive, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A \cup B) = P(A) + P(B)}

Example

Probability of stopping at first light is 0.4. Probability of stopping at second light is 0.5. Probability of stopping at at least one light is 0.6:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} P(A) & = 0.4\\ P(B) & = 0.5\\ P(A \cup B) & = 0.6\\ P(A \cap B) & = 0.3\\ P(A \cup B') & = 0.1\end{align}}


Equally Likely Outcomes

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is number of possible outcomes and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is event, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A)=\frac{N(A)}{N}}

Conditional Probability

Probability of event Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} given event Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} has occurred:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)}}

Lecture 4

Lecture 4 Notes

Thursday, January 27, 2011

Sample Warm-up Problem

Jane is taking a flight. Chance that she will sit in front row is .4; chance that her flights will be delayed is .3; chance that she willl sit in front row and flight is on time is .2:

(Draw a venn diagram)

We know:

  • A: sits in front row = .4
  • B: flight is delayed = .3
  • A∩B': front and not delayed = .2

We can find:

  1. P(A ∩ B) = P(A) − P(A∩B') = .2
  2. P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = .5
  3. P(A' ∩ B') = P( (A ∪ B)' ) = 1 - P(A ∪ B) = .5
  4. P(B' | A) = P(B' ∩ A) ÷ P(A) = 0.2 ÷ 0.4 = 0.5
  5. P(A' | B) = P(A' ∩ B) ÷ P(B) = 0.1 ÷ 0.3 = 1/3


Independent Events

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}P(A \cap B) &= P(A) \times P(B)\\P(A\cap B\cap \ldots \cap Z) &= P(A) \times P(B) \times \ldots \times P(Z)\end{align}}

B has no effect on A when independent:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A|B) = P(A)}

  • A: student is Male
  • B: student is an engineer
  • P(A) = .513
  • P(B) = .518
  • P(A ∩ B) = .414

.513 × .518 ≠ .414 ∴ Not independent.

Mutually Exhaustive

NOT TO BE CONFUSED WITH MUTUALLY EXCLUSIVE

(Mutually Exclusive means that P(A ∩ B) = 0)

Mutually Exhaustive means that P(A ∪ B) = Sample Space

Events that are Mutually Exclusive AND Mutually Exhaustive, then the sample space can be partitioned based on each event.

For any event D that is partitioned by Mutually exhaustive and exclusive events A, B, and C, then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(D) = P(A \cap D) + P(B \cap D) + P(C \cap D)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A \cap D) = P(D|A)\times P(A)}

…which brings us to…

Bayes' Theorem

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1, \ldots ,A_n} are mutually exclusive and exhaustive events:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A_i|B) = \frac{P(A_i \cap B)}{P(B)} = \frac{P(B|A_i)P(A_i)}{\sum_{j=1}^n P(B|A_j)P(A_j)}}

Example 1

Suppose a colon cancer test is 95% accurate (i.e. if a patient has colon cancer, the test will detect it 95% of the time)

If there is no cancer, the test will give a false positive 1% of the time.

If 5% of the population has colon cancer, what is the probability that a randomly selected patient does not have cancer given a positive test result.

Events:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1} : Patient does not have cancer
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2 = A_1'} : Patient has cancer
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} : Test returns positive

A1 and A2 are mutually exclusive and mutually exhaustive, therefore:

We know:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A_2)=0.05} 5% of population has cancer, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A_1) = 1-P(A_2)=0.95}
Since the test is 95% accurate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(B|A_2)=0.95}
Since the test returns a false positive 1% of the time, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(B|A_1)=0.01}

We want to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A_1|B)}


Solution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A_1|B)=\frac{P(B|A_1)P(A_1)}{P(B|A_1)P(A_1) + P(B|A_2)P(A_2)} = \frac{0.01 \times 0.95}{0.01 \times 0.95 + 0.95 \times 0.05}=0.167}

Example 2

Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn’t rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Marie’s wedding?

  • Let A be the event that it rains
  • Let B be the event that the weatherman predicts it will rain
Mutually Exhaustive and Exclusive Events

The two events A and A' are mutually exclusive and exhaustive.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} P(A) & = 5/365 \\ P(B|A) & = \tfrac{P(A \cap B)}{P(A)} = 0.9 \\ P(B|A') & = \tfrac{P(A' \cap B)}{P(A')} = \tfrac{P(A' \cap B)}{1-P(A)} = 0.1 \\ P(A|B) & = \tfrac{P(B \cap A)}{P(B)} = ? \end{align}}

Using the given information, we can solve for P(A ∩ B) and P(B):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A \cap B) = P(B|A) \times P(A) = 0.9 \times 5/365}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(B) = P(A \cap B) + P(A' \cap B) = 0.9 \times 5/365 + 0.1 \times (1-5/365)}

Plug these values into the equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(A|B) = \frac{0.9 \times 5/365}{0.9 \times 3/365 + 0.1 \times (1-5/365)} \approx 0.111}
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