Transcript Probability - St. Joseph School District

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.
2.
3.
4.
All 3 children are girls
There is one girl
There is at least one girl
There is at most one girl
Answers
Find each probability:
1.
2.
3.
4.
All 3 children are girls
There is one girl
There is at least one girl
There is at most one girl
1/8
3/8
7/8
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)