AP Statistics Chapter 6.3 Day 2
LEQ: How can tables be used to find conditional probabilities?
The probability we assign to an event can change if we know that some other event has occurred.
Age
Groups
Let’s examine this table from our chapter 4 test.
Urban
110
240
53
Under 25
25-50
Over 50
Totals
403
Localities of Residence
Suburban
Rural
150
65
220
75
112
58
482
198
Totals
325
535
223
1083
What is the probability of selecting an urban dweller? ___________________________________
What is the probability of selecting a person 25 – 50? ____________________________________
What is the probability of selecting an urban dweller that is 25 – 50? _____________________________
What is the probability of selecting an urban dweller given that the person is 25 – 50? ________________
When given a condition—this has the effect of reducing the size of the sample space, and therefore the
value of the denominator in a probability fraction.
Examples:
Consider rolling two dice and observing the sum. From our previous experiments, how many possible
outcomes are there?________________ This is true when there are NO special conditions in place.
If there is a condition – P(sum = 8 given that one die shows a 3)
Let’s look at these possibilities: the given condition is a three is showing – what are those dice
combinations?
(3, 1),
So, there are a total of ________________. This is our bottom number. How many of these have a sum
of 8?____________
So the P(sum of 8 given that one die shows a 3) = _________________.
The notation P(A|B) reads
“the probability of event A given that event B has already occurred”.
Let’s go back to our table –
Find the following probabilities using the table:
P(Suburbanite) = ______________________
P(under 25) = ________________________
P(Suburbanite | under 25) = _________________
P(Suburbanite and under 25) = ________________
What do we know about P(A and B)? If the events are independent, then ___________________________
Does P(Suburbanite) x P(under 25) = P(Suburbanite and under 25)?_____________
We need a new rule when events are not independent: Look at the values, which 2 have a product = to
P(Suburbanite and under 25)?
OUR new rule for any event: _______________________________________________________
In a very simple experiment: Let A = drawing a King Let B = drawing another King
Find the probability of drawing another king if you drew a king already P(King | King) =
Find the P(both cards are kings) =
To Summarize:
The joint probability that events A and B both happen can be found by
P(A  B)  P(A)  P(B|A)
The multiplication rule is just common sense made formal!
Example:
29% of Internet users download music files, and 67% of downloaders say don’t care if the music is
copyrighted. So the percent of Internet users who download music and don’t care about copyright is . . .
P(A  B)  P(A)  P(B|A) What if we know P(A) and we know P(A and B), the formula can be rearranged
to produce the definition of conditional probability:
Assignment: p. 446 #6.71, 74 - 77