Notes for math 141
Section 7.5-7.6
Finite Mathematics
7.5: Conditional Probability
Example 1: A survey is done of people making purchases at a gas station: buy drink
(D) no drink (Dc) Total
Buy drink(D)
Buy Gas (G)
20
c
No Gas (G )
10
Total
30
No drink(Dc )
15
5
20
Total
35
15
50
a) What is the probability that a person buys a drink?
b) What is the probability that a person doesnt buy a drink?
c) What is the probability that a person buys gas and a drink?
d) What is the probability that a person buys gas but not a drink?
e) What is the probability that a person who buys a drink also buys gas?
f) What is the probability that a person who doesnt buy a drink buys gas?
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Definition: If E and F are events in an experiment and P (E) ≠ 0, then the conditional probability that the event F will occur given that the event E has already
occurred is
P (E ∩ F )
P (F ∣E) =
.
P (E)
Definition: The Product Rule is found by rearranging the above formula as follows:
P (E ∩ F ) = P (E)P (F ∣E).
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Example 2: Lets use a tree diagram to help us understand the product rule:
Example 3: At a party, 13 of the guest are women. Seventy-five percent of the women
wore sandals and 40 % of the men wore sandals.
a) What is the probability that a person chosen at random at the party is a man
wearing sandals?
b) What is the probability that a person chosen at random is wearing sandals?
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Example 4: Consider drawing 3 cards from a standard deck of 52 cards without
replacement.
a) What is the probability that the 3 cards are hearts?
b) What is the probability that the third card drawn is a heart given the first two
cards are hearts?
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Independent events
Definition: If A and B are independent events, then P (A) = P (A∣B) and P (B) =
P (B∣A). Thus, two events A and B are independent if and only if
P (A ∩ B) = P (A) ⋅ P (B).
Example 5: A medical experiment showed the probability that a new medicine was
effective was .75, the probability of a certain side effect was .4, and the probability of
both occuring was .3. Are the events independent?
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Example 6: Meghan and Natalie go to Freebirds. After ordering their food they get
to roll a pair of fair dice for a chance to get their meal for free. Each die has six sides
with one of the sides having a backwards F on it. If both of the dice land on the
backwards F, you win. What is the probability that at least one of the girls wins?
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
7.7 Bayes Theorem
Example 7: If we are given information about P (F ∣E), can we find P (E∣F )?
Definition: The above formula is known as Bayes Theorem.
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Example 8: We are to choose a marble from a cup or a bowl. We flip a fair coin to
decide whether to choose from the cup or the bowl. The bowl contains 1 red and 2
green marbles. The cup contains 3 red and 2 green marbles. What is the probability
that a marble came from the bowl given that it is red?
Example 9: A crate contains 7 basketballs and 4 footballs. A bag contains 4 basketballs and 2 footballs. A ball is drawn at random from the crate and put in the
bag. A ball is then drawn from the bag. Given that a basketball was chosen from the
bag, what is the probability that a football was drawn from the crate?
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Example 10: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a face card given
that the second card is an ace?
Example 11:Complete the following tree diagram and use it to answer the following
questions:
a) Find P (E).
b) Find P (A ∪ D).
c) Find P (B ∩ E).
d) Are E and B independent events?
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
8.1:Random Variables and Histograms
Definition: A random variable is a rule that assigns a number to each outcome of
an experiment.
Example 1: Suppose we toss a coin three times. Then we could define the random variable X to represent the number of times we get tails.
Example 2: Suppose we roll a die until a 5 is facing up. Then we could define
the random variable Y to represent the number of times we rolled the die.
Example 3: Suppose a flashlight is left on until the battery runs out. Then we
could define the random variable Z to represent the amount of time that passed.
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Types of Random Variables
1. A finite discrete random variable is one which can only take on a limited number
of values that can be listed.
2. A infinite discrete random variable is one which can take on an unlimited
number of values that can be listed in some sort of sequence.
3. A continuous random variable is one which takes on any of the infinite number
of values in some interval of real numbers. (i.e. usually measurements)
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Example 4: Classify each of the following random variables as finite discrete, infinite
discrete, or continuous and list/describe the possible values for the random variable.
a) A drawer contains 15 red pens, 8 blue pens, and 5 black pens. An experiment
consists of drawing pens out of the drawer with replacement until a red pen is drawn.
Let X represent the number of pens drawn.
b) An experiment consists of randomly selecting 3 pens without replacement from
a conveyor belt. Let Y represent the total weight (in ounces) of the selected pens.
c) An experiment consists of randomly selecting 3 pens with replacement from a
drawer that contains 15 red pens, 8 blue pens, and 5 black pens. Let Z represent the
number of red pens drawn.
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Example 5: Referring to Example 1, find the probability distribution of the random
variable X.
Example 6: An experiment consists of randomly selecting a sample of 3 grapes out
of a bowl that contains 20, of which 8 are rotten. Let X represent the number of
rotten grapes in the sample. Find the probability distribution of X.
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
Example 7: Blue Baker prides itself on having the best chocolate chips cookies in
town. To be sure each cookie has the right number of chocolate chips, a few cookies
are selected from each batch and the number of chocolate chips in each cookie is
counted. This is done for several days and the following results were found:
Number of Cookies
2 4 5
Number of Chocolate Chips 8 11 12
6 8
13 14
Identify the random variable X in this experiment and find the probability distribution
of X.
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Notes for math 141
Section 7.5-7.6
Finite Mathematics
We use histograms to represent the probability distributions of random variables.
We place the possible values for the random variable X on the horizontal axis. We
then center a bar around each x value and let its height be equal to the probability
of that x value
Example 8: Referring to Example 7,
a) Draw the histogram for the random variable X
b) Find P (X ≥ 12)
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