Download Name Math 2b/3a May 4, 2015 Probability Test Review Probability

Name ___________________________________
May 4, 2015
Math 2b/3a
Probability Test Review
Probability Review Part 1:
Counting, Pascal’s Triangle, and the Binomial Theorem
Review of Counting Question Types:
Type 1: You are asked to count the number of distinct ways of arranging a group of objects.
a. How many different ways are there of arranging a group of different things in a
row/line/list?
Example: Mrs. Runge is planting flowers in a window box outside her kitchen window.
She plants 8 flowers, each of which is a different color. How many different ways are
there of arranging the flowers?
b. How many different ways are there of
arranging a group of things in a
row/line/list when some are repeated?
c. How many different ways are there of
arranging a group of different things in a
circle?
Example: Mrs. Runge is planting flowers in
a window box outside her kitchen window.
She plants 8 flowers: 3 red, 2 yellow, and 3
white. How many distinct ways are there of
arranging the flowers?
Example: Mrs. Runge is planting a circular
garden. She plants 8 flowers around the
edge of the garden, each of which are
different colors. How many distinct ways
are there of arranging the flowers?
8!
= 560
3!2!3!
8!
= 5040
8
d. How many different ways are there of arranging a group of objects in a circle when
some are repeated?
Example: Mrs. Runge is planting a circular garden. She plants 8 flowers around the edge
of the garden: 3 red, 2 yellow, and 3 white. How many distinct ways are there of
arranging the flowers?
8!
= 70
3!2!3!8
Name ___________________________________
May 4, 2015
Math 2b/3a
Probability Test Review
Type 2: You are asked to count the number of ways of picking a few things out of a larger group.
a. How many ways are there of picking a
few things out of a group, when the order
they are selected matters?
b. How many ways are there of picking a
few things out of a group, when the order
they are selected does not matter?
Example: There are 10 students running for
class office. The student who gets the most
votes becomes president, second most
becomes vice-present, third becomes
treasurer and fourth becomes secretary.
How many different groups of class
officers are there?
Example: There are 10 students running for
class office. The four students who get the
most votes are elected as class officers, and
they each share the different
responsibilities. How many different
groups of class officers are there?
10P4
= 5040
10C4
=
10!
= 210
4!6!
3. Write down the first 7 rows of Pascal’s Triangle.
4. Find each of the following combination numbers using Pascal’s Triangle. Remember this is
the weird notation from the book for 6C2 …
æ
ö
æ
ö
æ
ö
a. ç 6 ÷
b. ç 5 ÷
c. ç 3 ÷
è 2 ø
è 0 ø
è 1 ø
5. Expand the following using the Binomial Theorem. Completely simplify your answer.
(x + 2)6
6. Find the number of different ways of rearranging the letters in the following names.
a. PAUL
b. LENNA
c. ACADIA
d. NIKKI
7. There are 28 plants in the window box outside of Ranc’s. There are 10 with purple flowers,
10 with yellow flowers, and 8 with orange flowers. How many different ways of arranging
the plants are there? For this problem, you do not need to multiply/divide to get a single
number as your answer.
8. In the courtyard of Mrs. Runge’s apartment building, there is a circular patch of grass with
tulips planted along the edge. There are 24 red tulips and 12 yellow tulips. Find the number
of different ways the tulips could be arranged. For this problem, you do not need to
multiply/divide to get a single number as your answer.
Name ___________________________________
May 4, 2015
Math 2b/3a
Probability Test Review
9. a. There are 6 Lexpress busses in line in the Center. There is one of each number bus.
b. Now suppose there are three #1 busses, one #2 bus, one #3 bus, and one #4 bus. How
many different ways could the busses line up?
10. a. Suppose for next year there are 5 sections of Math 2B/3A, and there are 20 math
teachers. How many different ways are there of just choosing a teaching team of 5
different teachers?
b. Now suppose instead of just choosing a teaching team, the department head needs to also
make scheduling assignments. He first chooses a teacher to teach an A block section, then
he chooses a second teacher to teach a B block section, up to a fifth teacher to teach an E
block section. How many different ways of making teaching assignments are there?
11.
You want to make a bouquet of five different types of flowers from the 12 flowers
available. How many bouquets could you make?
Name ___________________________________
May 4, 2015
Math 2b/3a
Probability Test Review
Probability Review Part 2:
Mutually Exclusive, Independent, and Conditional Probabilities & Expected Value
Two events A and B are mutually exclusive if they do not share any outcomes in the sample
space.
Example: Rolling a 6 and rolling a 3 when you toss a die one time.
Non-Example: getting a heads and getting a tails when you flip a coin two times.
Two events A and B are independent if the result from one event has no effect on the other. If A
and B are independent, then P(A and B) = P(A) * P(B).
Example: rolling a 6 and then rolling a 3.
Non-Example: being female and being a mother
Important Notes:

If two events A and B are mutually exclusive, then P(A or B) = P(A) + P (B).

If two events A and B are independent, then P(A and B) = P(A) * P (B).__.

The probability of an event A not happening is P(not A) = 1 – P(A).

If two events A and B are not mutually exclusive, then
P(A and B) = P(A) + P(B) – P(A and B)
Expected Value finds what one should expect to happen on average (after many trials) in a
given situation. You must know the outcomes & their values, the probabilities for each outcome
in order to find the expected value. A value greater than 0 is favorable to the person whose
perspective is represented in the outcome/value column. If the value is negative, then it is
unfavorable and a value of 0 means there is no net gain or loss and this is considered to be fair.
Name ___________________________________
May 4, 2015
Math 2b/3a
Probability Test Review
Practice Problems:
12. A bag contains 7 blue marbles, 5 red marbles, and 2 white marbles. You select two marbles
without replacement.
a. Find the probability that the two marbles are the same colors. Hint: There are three
different ways
this can happen.
b. Find the probability that the two marbles are different colors. Hint: This should be pretty
quick if you use what you found in part a.
13. Suppose you randomly select an LHS student. Use the following events and their
probabilities to find the probabilities listed below, or explain why you can’t.
S: student is a sophomore, P(S) = 0.23 F: student is a freshman, P(F) = 0.27
B: student has blue, P(B) = 0.17
E: student is taking Earth Science, P(E) = 0.28
M: student’s favorite subject is math, P(M) = 0.90 
a.
b.
c.
d.
e.
f.
P(F and M)
P(F or S)
P(B and E)
P(M|S)
P(S|F)
P(M or E)
14. You play a carnival game where you roll a number cube and flip a coin. The coin has the
number 2 on 2 on one side and 6 on the other side. Your score is the sum of the values that
appear on the number cube and the coin flip.
a. Write out the sample space for this experiment.
b. You win a stuffed animal if you score 11 points or more. Find the probability of winning.
15. Consider the number of students who are late to a class the first block of the day. Use the
table to the right to find the expected number of students late to class. Explain what this
expected value means.
Number of
students late
Probability
0
1
2
3
4
5
6
7
8
9
0.01
0.02
0.03
0.04
0.10
0.20
0.35
0.15
0.07
0.03
16. If a voter is chosen at random from a large group of registered voters, the probability that the
voter is a registered Democrat is 0.46 and the probability that the voter is a registered
Republican is 0.45. (A voter is not allowed to be registered in both parties.)
a. What is the probability that a randomly chosen voter is not a Democrat?
b. What is the probability that a randomly chosen voter is a Democrat or a Republican?
c. Voters that are neither Democrat nor Republican are called Independent.
What is the probability that a randomly chosen voter is an Independent?
Name ___________________________________
May 4, 2015
Math 2b/3a
Probability Test Review
17. On any given day in May, suppose you wear jeans to school with probability 0.6 and you
wear shorts to school with probability 0.3. You wear something else, like sweatpants, with
probability 0.1. Let A be the event that you wear jeans, let B be the event that you wear
shorts, and let C be the event that you wear something else. Find each of the following
probabilities.
a.
b.
c.
d.
e.
f.
g.
P(A or B)
P(A and C)
P(A|B)
P(not B)
P(neither A nor B)
P(A or B or C)
P(A and B and C)
18. Here are the actual probabilities for the problem about taking 5 cell phones and returning
them at random. The table at the right shows the probabilities for how many people get their
own phones back.
a. Why do all the probabilities have 120 as
their denominator? Hint: The number
120 comes from a counting problem.
number of people who get
back their own phone
n=0
n=1
n=2
n=3
n=4
n=5
probability
44
120
45
120
20
120
10
120
0
1
120
b. On average, how many people will get their own phones back? In other words, calculate
the expected value for the number of people who get back their own phone.
19. In a drawer there are 7 green tennis balls (G) and 3 pink tennis balls (P). Suppose that one
ball is taken from the drawer, and then (without replacing the first one) a second ball is taken.
a. Find all the outcomes for this experiment.
b. What is the probability that the first tennis ball is pink and the second tennis ball is pink?
c. What is the probability that both tennis balls are green?
d. What is the probability that the two tennis balls have different colors?
e. You make a deal with your sister that if you can pull two pink tennis balls out of the
drawer, she has to do your chores for 7 days, if one of the two is pink, she does your
chores for 3 days and if both are green you do her chores for 5 days. Who made the
better deal (Hint: what is the expected value)?