Conditional probability. Ordinary Probability You are dealt two cards from a deck. What is the probability the second card dealt is a Jack? We reason

Conditional probability

Ordinary Probability

You are dealt two cards from a deck. What is the probability the second card dealt is a Jack?

We reason that if the two cards have been delt, the probabiity that the first is a jacek and the probabiity that the second is a jack are identical. Since there are 4 jacks in the deck, we compute

P(second is jack) = 4/52 = 1/13.

Conditional Probability

You are dealt two cards from a deck. What is the probability of the second card was a jack given the first card was not a jack is called card dealt is a Jack?


You are dealt two cards from a deck. What is the probability that the second card was a jack given the first card was not a jack.


You are dealt two cards from a deck. What is the probability of the second card was a jack given the first card was not a jack is called card dealt is a Jack?

51

4:Answer

Reasoning

Once the first card has been delt, there are 51 cards remaining, and, since the first was NOT a jack, there are 4 jacks in the set of 51 cards.


The probability of drawing jack given the first card was not a jack is called conditional probability. A key words to look for is “given that.”We will use the notation:

P(second a jack | first not a jack) = 4/51

General Conditional Probability

The probability that the event A occurs, given that B occurs is denoted:

This is read the probability of A given B.

).|( BAP


How would we draw the event A given B?

A B A

and B



Since we know B has occurred, we ignore everything else.

A B A

and B




A B A

and B

Conditional ProbabilityHow would we draw the event A given B?


With some thought this tells us:

B A

and B

)(

) and ()|(

BP

BAPBAP

Additional notes

In the case of a equi-probability space, we can reason that, since we know the outcome is in B, we can use the set B as our reduced sample space. The probability P(A|B) can then be computed as the number of points in A∩B as a fraction of the number of points in B.

Continuing…

This gives

Dividing top and bottom by the numbers of

points in the original sample space S:

Example from well contamination

Bars show percents

Below Limit Detect

MTBE-Detect

0%

25%

50%

75%

79% 21%

Private Public

Below Limit Detect

MTBE-Detect

60% 40%

Of private wells, 21% are contaminated. Therefore

P(C|Private)=0.21

Of public wells, 40% are contaminated. Therefore

P(C|Public)=0.40

Law of conditional probability

𝑃 ( 𝐴|𝐵 )= 𝑃 (𝐴∩𝐵)𝑃 (𝐵)

Multiplication Rule

𝑃 ( 𝐴)=𝑃 (𝐵 )𝑃 (𝐴∨𝐵)

Example

A local union has 8 members, 2 of whom are women. Two are chosen by a lottery to represent the union.

Example


a) What is the probability they are both male?

Example



M)|P(M(M):Answer P

Example



536.07

5

8

6:Answer

Example


b) What is the probability they are both female?

Example


b) What is the probability they are both female?

036.07

1

8

2F)|P(F*P(F):Answer

Example

Find the probability of selecting an all male jury from a group of 30 jurors, 21 of whom are men.

Example

Find the probability of selecting an all male jury from a group of 30 jurors, 21 of whom are men.

Solution:

P(12 M) =P(M)*P(M|M)*P(M|MM) * ….

= 21/30 * 20/29 * 19/28 * 18/27 * … 10/19

= 0.00340

Independent Events

Two events A and B are independent if the occurrence of one does not affect the probability of the other.

Independent Events


Two events A and B are independent then P(A|B) = P(A).

Independent Events


Two events A and B are independent then P(A|B) = P(A).

Two events which are not independent are dependent.

Multiplication Rule

Multiplication Rule:

For any pair of events:

P(A and B) = P(A) * P(B|A)

Multiplication Rule




For any pair of independent events:

P(A and B) = P(A) * P(B)

Multiplication Rule



For any pair of independent events:

P(A and B) = P(A) * P(B)

If P(A and B) = P(A) * P(B), then A and B are independent.

Multiplication Rule


P(A and B) = P(A) * P(B) if A and B are independent.

P(A and B) = P(B) * P(A|B) if A and B are dependent.

Note: The multiplication rule extends to several events: P(A and B and C) =P(C)*P(B|C)*P(A|BC)

ExampleA study of 24 mice has classified the mice by two categories

Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour

A study of 24 mice has classified the mice by two categories

a) What is the probability that a randomly selected mouse has white fur?

b) What is the probability it has black eyes given that it has black fur?

c) Find pairs of mutually exclusive and independent events.

Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour


a) What is the probability that a randomly selected mouse has white fur? 12/24=0.5

b) What is the probability it has black eyes given that it has black fur?


Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour


a) What is the probability that a randomly selected mouse has white fur? 12/24=0.5

b) What is the probability it has black eyes given that it has black fur? 1/4=0.25

c) Find pairs of independent events.

Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour

b) What is the probability it has black eyes given that it has black fur? 1/4=0.25


IND: White Fur and Red Eyes; Black Fur and Red Eyes

Black White Grey

Eye Colour

Red Eyes 3 5 2

Black Eyes 1 7 6

Fur Colour

Descriptive Phrases

Descriptive Phrases require special care!

– At most– At least– No more than– No less than

Review

• Conditional Probabilities• Independent events• Multiplication Rule• Tree Diagrams

Documents

Conditional probability. Ordinary Probability You are dealt two cards from a deck. What is the probability the second card dealt is a Jack? We reason