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16.2 Probability of Events
Occurring Together
Definitions
• Conditional Probability
Conditional probability is the probability of
an event given that you know another
event has occurred.
Notation
Notation
Read As
P(A|B) “P of A given B”
Means
The probability
that event A
occurs, given
that event B
has occurred.
Guided Practice
Let’s find the following probabilities given I
randomly choose a person in the classroom.
P(A girl)
P(A girl | A person wearing pink)
P(A girl | plays softball)
P(A girl | plays lacrosse)
P(Plays lacrosse | A girl)
Guided Practice
There is a jar of 3 blue marbles and 5 yellow
marbles without replacement. Find:
P(BY) (means blue, then yellow)
P(YB) (means yellow, then blue)
P(YY)
P(BB)
P(2nd is B|1st is Y)
P(2nd is Y|1st is Y)
P(2nd is Y|1st is B)
Conditional Probability
• The probability of event A occurring, given
that event B has occurred is:
P(A|B) = P(A and B)
P(B)
“Given” goes in the denominator!
Note: P(B) must not equal 0.
Example 2.
• Draw a card at random from a standard deck.
Event A is you draw a queen. Event B is you
draw a red card. What is the probability that
you draw a queen, given that the card is red?
4
26
2
P( A) 
, P( B)  , P( A  B) 
52
52
52
2
P( A  B) 52 2 52 2
P( A | B) 

 

26 52 26 26
P( B)
52
General Multiplication Rule
• The probability of two events both
occurring is the probability of the first times
the probability of the second, given that
the first has occurred:
P(A and B) = P(A) ∙ P(B|A)
as well as
P(A and B) = P(B) ∙ P(A|B)
Definitions
• Independent Events
Two events are said to be independent if
knowing the outcome of one event does
not change the probability of the other
event occurring.
That is, P(B|A) = P(B) and
P(A|B) = P(A)
Examples of Independence
• The probabilities of choosing a red card
and queen from a standard deck of cards
are independent.
• The probabilities of choosing a girl at
random from the class and choosing a
softball player are not independent.
Multiplication Rule for
Independent Events
• The probability of two independent
events both occurring is the product of
their individual probabilities:
P(A and B) = P(A) ∙ P(B)
• Note: If this multiplication rule holds, then
two events are independent.
• Ex. Find the P(Heads and Rolling a 3)
Guided Practice
You draw one card from a standard deck.
Event A is the card is a heart. Event B is the
card is face card.
1. Are these events independent? How can
you know?
2. State an Event in this sample space that
would NOT be independent from Event A.