Honors Mathematical Analysis
Section 1.6 Problems: Expected Value
Q Problems
Q1.) Simplify: (x + 5)2 (Be careful!)
Q2.) Solve for x by factoring: x2 -2x – 8 = 0
Q3.) Write the quadratic formula:
Q4.) Solve for x using the quadratic formula: x2 – 9x = -18
(𝑥−3)
Q5.) Simplify: (𝑥−3)(𝑥+5)
Q6.) List four methods of proving triangles congruent.
Q7.) Find the area of a 45-45-90 triangle with an 8√2 inch hypotenuse.
Q8.) Determine the length of the long leg of a 30-60-90 triangle with a 10 cm hypotenuse.
Q9.) If an angle measures yº, write an expression representing the measure of its supplement.
Q10.) If ∠𝐴 ≅ ∠𝐵 and ∠𝐵 ≅ ∠𝐶, then ∠𝐴 ≅ ∠𝐶. This illustrates what property of congruence?
1.) Students at a school carnival are awarded points for winning games. At the end of the evening, they can trade in
their points for prizes. For a particular game, students start out with 50 points and roll a single die. The payoffs are:
 Roll a 6: Win 100 points (and get your 50 points back)
 Roll a 2 or 4: Win 10 points (and get your 50 points back)
 Roll an odd: Win nothing (lose your 50 points)
What are the expected winnings (or losings) for the game?
2.) When a uranium atom splits during fission, it releases 0, 1, 2, 3, or 4 neutrons. Let P(x) be the probability that x
neutrons are released. Assume the probability distribution is:
x
0
1
2
3
4
P(x)
0.05
0.2
0.25
0.4
0.1
What is the mathematically expected number of neutrons released per fission?
3. An archer has the following probabilities of hitting the various rings
shown in the table below:
Color
Probability
Points
Gold
0.20
9
Red
0.36
7
Blue
0.23
5
Black
0.14
3
White
0.07
1
A) What is her mathematically expected number of points on any one shot?
B) If she shoots 48 arrows, what would her expected score be?
4.) A car dealership offers its sales staff two bonus options during a week-long sale:
Option A: Receive a $100 bonus for each vehicle sold
Option B: Receive a $2000 bonus for selling four minivans, two station wagons, one hybrid, and one sedan
The probability of selling four minivans is 50%, two station wagons is 70%, a hybrid is 80%, and a sedan is 90%.
A) What is the mathematical expectation for a bonus for a salesperson choosing Option A and selling the required four
minivans, two station wagons, one hybrid, and one sedan?
B) What is the probability of being able to achieve the requirement for Option B?
C) What is the mathematical expectation for a bonus for Option B?
D) Based on your calculation, which option would you choose if you worked for the dealership?
5. A game room has a “Claw” machine that allows players to attempt to grab a prize. The cost to play the game is $1. A
player can expect to win a $6 toy 2.5% of the time, a $3 toy 5% of the time, or a $1.50 bag of candy 20% of the time.
A) What is a player’s mathematical expectation for any given game? (Remember to subtract the cost of playing the game
from any winnings!)
B) If a vendor wants to make at least $0.60 for each game played, what should be charged?
6. The basketball toss at an amusement park costs 50 cents to play. To play, you shoot three balls. If you make no
baskets, you win nothing (and lose your 50 cents). If you make exactly one basket, you win a key chain worth 5 cents. If
you make exactly two of three baskets, you win a 60 cent stuffed animal. If you make all three shots, you win a doll
worth $2.50. Since the hoop is small, the probability of making any shot is 30%.
A) What is the probability of making NONE of the shots?
B) What is the probability of making ALL of the shots?
C) What is the probability of making exactly one shot? (Keep in mind that this can be done in several different orders!)
D) What is the probability of making exactly two shots?
E) What is the mathematical expectation for the game? (Remember to subtract the cost of playing the game from any
prize values won!)
7. Insurance companies rely on calculations based on probability. Actuaries are mathematicians who calculate prices
charged by insurance companies based on statistics. Suppose an insurance company has an accident/illness policy that
pays $500 if you get sick during any given year, $1000 if you have an accident, and $6000 if you get sick AND have an
accident. The premium (payment) for this policy is $100 per year. According to statistics, your probability of becoming
ill in any one year is 0.05 and your probability of having an accident is 0.03.
A) Find the probability of:
i) Becoming ill AND having an accident
ii) Becoming ill and NOT having an accident
iii) Not becoming ill, but having an accident
iv) Not becoming ill and not having an accident
B) What is the mathematical expectation for this policy?
C) An insurance policy is actuarially sound if the company can expect a profit from it. Based on the given information, is
this policy actuarially sound?