Probability vs Statistics



Probability vs Statistics

Probability: Predicts nature of samples from knowledge of population

probability of rolling “7” with two fair dice

probability of drawing a royal flush

probability of winning a “pick 3” game

Statistics: Predict nature of population from knowledge of samples

forecasting demand from past data

evaluating effectiveness of a new drug

determining average yield of a chip manufacturing process

Question: Suppose you toss a coin 7 times and it comes up “Heads” every time. What do you think is likely to happen on the 8th throw?

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Probability Example

Experiment: Roll 2 dice

Sample space: 36 possible outcomes (equally likely)

|# on 1st Die |

| | |1 |2 |3 |4 |5 |6 |

| |1 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|# on |2 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|2nd |3 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|Die |4 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |5 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |6 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

Each cell in the table represents a possible outcome.

Each outcome is equally likely, with probability 1/36.

Events are collections of outcomes, or cells.

Example 1

Event A: {First die thrown shows a ‘3’}

|# on 1st Die |

| | |1 |2 |3 |4 |5 |6 |

| |1 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|# on |2 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|2nd |3 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|Die |4 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |5 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |6 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

A = {(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)}

P{A} = 1/36 + 1/36 + ... + 1/36 = 1/6

Example 2

Event B: {Sum of two dice = ‘7’}

|# on 1st Die |

| | |1 |2 |3 |4 |5 |6 |

| |1 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|# on |2 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|2nd |3 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|Die |4 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |5 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |6 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

B = {(1,6) (2,5) (3,4) (4,3) (5,2) (6,1)}

P{B} = 1/36 + 1/36 + ... + 1/36 = 1/6

Example 3

Event A U B: {First die thrown shows a ‘3’ or sum of dice = ‘7’}

|# on 1st Die |

| | |1 |2 |3 |4 |5 |6 |

| |1 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|# on |2 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|2nd |3 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|Die |4 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |5 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |6 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

It is not true that: P(A U B} = P{A} + P{B}

In this case, P{A U B} = 11/36 < 1/6 + 1/6

Note: Be Careful not to double count! The outcome (3,4) is in both events: it gets counted only once.

Fact: In general,

P{A U B} = P{A} + P{B} - P{A ( B}, so in this case

P{A U B} = 1/6 + 1/6 - 1/36 = 11/36.

Conditional Probability

Definition: (Baye’s Rule)

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Example:

A = {sum of two dice = ‘7’}

B = {first die = ‘3’}

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|# on 1st Die |

| | |1 |2 |3 |4 |5 |6 |

| |1 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|# on |2 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|2nd |3 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

|Die |4 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |5 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

| |6 |1/36 |1/36 |1/36 |1/36 |1/36 |1/36 |

Independence

Recall that:

P{A ( B} = P{A | B}P{B}.

If

P{A ( B} = P{A } P{B},

then A and B are said to be independent.

Example: Tossing a ‘fair’ two-sided coin

A = {1st two throws are heads}

B = {1st three throws are heads}

C = {Third throw is a head}

P{A} = 1/4 P{B} = 1/8 P{C} = 1/2

P{A | B} = 1 P{B | C} = 1/4 P{C | A} = 1/2

P{A ( B} = P{A | B} P{B} = 1/8 ( P{A}P{B}, so A and B are NOT independent

P{B ( C} = P{B | C} P{C} = 1/8 ( P{B}P{C}, so B and C are NOT independent

P{A ( C} = P{C | A} P{A} = 1/8 = P{A} P{C}, so A and C ARE independent

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