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’}
[pic]
|# 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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