Probability

Part II

Tree Diagram

Used to show all of the possible outcomes of an experiment

Example

A couple plans on having 3 children. Assuming that the births are single births, make a tree diagram.

Solution

• There are 8 birth orders.

• BBB• BBG• BGB• BGG• GBB• GBG• GGB• GGG

Questions

Find each probability:

1. All 3 children are girls

2. There is one girl

3. There is at least one girl

4. There is at most one girl

Answers

Find each probability:

1. All 3 children are girls 1/8

2. There is one girl 3/8

3. There is at least one girl 7/8

4. There is at most one girl 4/8

Independent events

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

Examples

• Rolling a pair of dice

• Tossing 2 coins

• Drawing 2 cards from a deck if the first card is replaced before the second card is drawn

Dependent events

Two events, A and B, are dependent if the occurrence of one event does affect the probability of the occurrence of the other.

Examples

• Drawing 2 cards from a deck of cards if the first card is not replaced before drawing the second card.

• Note: Without replacement is a clue that the events will be dependent.

Multiplication Rule

Independent Events

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

Dependent events

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

P(B|A) means probability of B assuming that A has happened. It is called a conditional probability.

Example

• A die is rolled twice. What is the probability that the first roll is an even number and the second roll is a number greater than 4?

Solution

• These are independent events.

• P(even number) = 3/6

• P(number > 4) = 2/6

• P(A and B) = 3/6 * 2/6

= 6/36

= 1/6

Example

• Two cards are drawn from a deck of cards. What is the probability that both are Kings, if

a. The first card is replaced before drawing the second card

b. The first card is not replaced before drawing the second card

Solution

• There are 4 Kings in a deck of 52 cards.

• With replacement:

P = 4/52 * 4/52

= 0.006

• Without replacement

P = 4/52 * 3 / 52

= 0.005

Tables to find conditional probabilities

• A sample of 1000 people was obtained. There were 500 men and 500 women. Of the men, 63 were left handed. Of the women, 50 were left handed.

Men Women Total

Left handed 63 50 113

Right handed 437 450 887

Total 500 500 1000

Example

• What is the probability that the person is a male given the person is right handed?

Example

• What is the probability that the person is a male given the person is right handed?

• Solution: There were 887 right handed people. Of these, 437 were men.

• P(M|RH) = 437/887

= 0.493

Example

• What is the probability that person is right handed, given the person is male?

Example

• What is the probability that person is right handed, given the person is male?

• Solution: There were 500 males. Of these, 437 were right handed.

• P(RH|M) = 437/500

= 0.874

Testing independence for a table

• Two events will be independent if

P(B|A) = P(B)

Example

• Are the events “male” and “right handed” independent or dependent?

Solution

• P(male) = 0.500

• P(male|right handed) = 0.493

These are not equal, so the 2 events are dependent.

Note: You could also see if P(right handed) = P(right handed|male)