c
Math 166, Fall 2015, Robert
Williams
1.6- Conditional Probability
If E and F are two events in the same sample space S, the conditional probability that E occurs given that F has occured is defined to be
P (E|F ) =
By rearranging the above equation, we get the product rule
P (E ∩ F ) =
Example 1 In the card game blackjack, the numbered cards have the same value
as their number, a jack, queen, and king all have value of 10, and an ace has a
value of either 1 or 11. If a player draws a single card and tells you the value of
the card is at least 8, what is the probability that the card’s value is 10? Assume
that the player is counting an ace as 11.
1
c
Math 166, Fall 2015, Robert
Williams
Example 2 Your male coworker tells you that he has one sibling, but refuses
to tell you whether he has a sister or a brother. Assume that, when having
children, either sex is equally likely to be born.
1. Determine the probability that your coworker’s sibling is female if you
know that your coworker is the older child.
2. Determine the probability that your coworker’s sibling is female if you don’t
know which is older.
A finite stochastic process is a finite sequence of experiments in which the
.
probability of each outcome
Stochastic processes can be effectively described by probability trees.
Example 3 A bag of dice has one die 4 sides, one die with 6 sides, and four
dice with 12 sides. Assume that you are equally likely to pick any die out of the
bag. What is the probability of rolling no higher than a 4? Assuming that you
rolled no higher than a 4, what is the probability that you picked a 12-sided die?
2
c
Math 166, Fall 2015, Robert
Williams
Example 4 A sack of marbles contains 3 black marbles, 1 green marble, and 2
red marbles. You are asked to pull out marbles one at a time until you pull out
a red marble. What is the probability that you will pull out 3 marbles?
Indepent Events
We say that events E and F are independent if
, which can be written as
P (E|F ) = P (E)
and
P (F |E) = P (F )
The Indepent Events Theorem: Let E and F be two events with P (E) > 0
and P (F ) > 0. Then E and F are independent if and only if
P (E ∩ F ) = P (E)P (F )
We say that a set of events {E1 , E2 , . . . , En } is independent if, for any subset
of two or more of these events, the probability of the intersection of the events
is equal to the product of the probabilities of the events.
3
c
Math 166, Fall 2015, Robert
Williams
Example 5 A family has three children. Let B be the event “the oldest child
is a boy”, G be the event “the youngest child is a girl”, and V be the event “at
least one child is a boy and at least one child is a girl”. Which sets of (two or
more of ) these events are independent sets of events?
Example 6 The following data comes from records kept by the Texas Department of Transportation on the number of traffic accidents that involve possible
injury: In 2014, there were 162, 022 vehicle accidents of which 3, 189 were fatal
to at least one person involved. There was a total of 10, 536 crashes involving
a driver whose blood-alcohol level was above the legal limit, 925 of which were
fatal to at least one person involved. Let F be the event “fatal vehicle accident”
and D be the event “vehicle accident with driver above legal blood-alcohol level”.
Are F and D independent events?
4