AP STATS 12
5.3 – conditional Probability and independence
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CONDITIONAL PROBABILITY: The probability that one event happens given that another
event is already known to have happened. If we know that event A has happened, then
the probability of event B happens given that even A has happened is denoted by P (B A )
Let’s examine two different examples of conditional probability.
Ex 1: Suppose you select two cards from a shuffled deck of cards.
The first card is the Ace of Spades. What is the probability that the
second card will be a spade?
Ex 2: Suppose you spin the Magic Money Wheel two times. The first time
you land on $0. What is the probability that you will land on $0 again?
$5
$0
$10
$20
Q: Which previous example shows independent events?
INDEPENDENT EVENTS: Two events A and B are independent if the occurrence of the
first event has no effect on the chance that the second event will happen. Using
symbols, P (B A) = P (B ) .
THE MULTIPLICATION RULE for TWO or MORE EVENTS
The probability that events A and B both occur is P (AandB ) = P(A) × P (B A )
P (AandB ) is sometimes written as P (A ∩ B ) Ex 3: Find the probability that we draw two aces in a row in two draws from a regular
deck.
AP STATS 12
Ex 4: Using the table on U.S. Census Data, we had:
a) Find P(Black)
Hispanic Not Hispanic
Asian
0.000
0.036
Black
0.003
0.121
White 0.060
0.691
Other
0.062
0.027
b) Find P(Hispanic)
c) Find P(Black and Hispanic)
Conditional Probability Formula: (from the Multiplication Rule)
P ( AandB ) P(A ∩ B)
P(B A) =
=
.
P(A)
P(A)
d) Find P(Black Hispanic)
e) Find P(Hispanic Black)
Ex 5: (Using tree diagrams for conditional probability)
A company has two factories that make computer chips. Suppose 70% of the
chips come from factory 1 and 30% come from factory 2. In factory 1, 25% of the chips
are defective. In factory 2, 10% of the chips are defective.
a. Find P(a randomly selected chip is defective)
b. Find P(a chip comes from factory 1
chip is defective)
AP STATS 12
Ex 6: The probability that a randomly selected person has leukemia is 0.02%. A medical
screening test for leukemia is correct 99.2% of the time.
a) Draw a tree diagram illustrating this scenario.
b) Use the tree diagram to answer the following questions.
i.
Find the probability that a randomly selected person has leukemia
and tests positive.
ii.
Find the probability that a randomly selected person does not have
leukemia and tests positive.
iii.
Find the probability that a randomly selected person tests positive for
leukemia.
iv.
Find the probability that a randomly selected person tests positive,
given that they do not have leukemia.
v.
Find the probability that a randomly selected person has leukemia,
given that they’ve tested positive.
AP STATS 12
Some special cases...
The MULTIPLICATION RULE for INDEPENDENT Events
Since P (B A) = P (B ) and P (AandB ) = P(A) × P (B A) , then
The Special Case is P (AandB ) = P(A) × P (B )
The ADDITION RULE for NON-MUTUALLY EXCLUSIVE Events
If A and B are not mutually exclusive, then P(A or B) = P(A) + P(B) – P(A and B)
Ex 7: From Ex 6 above, use the above addition rule to find the probability that a
randomly chosen person has leukemia or tests positive.
Ex 8: A person facing MCL surgery knows that the operation fails 20% of the time. He
also knows that infection occurs in 1.5% of operations. The chance of failure and
infection is 1%. Find P(failure or infection).