2. What Are the Odds in Favor of Drawing a Spade and a Heart?
2. What are the odds in favor of drawing a spade and a heart?
Assuming you can draw them in either order…
26 valid cards on the first draw:
26/52
13 valid cards on the second draw:
13/51
Multiply:
26/52 * 13/51
= 13/102
Odds = p/(1-p) = 13/102 / (1 – 13/102) = 13/102 / 89/102 = 13/89
8. What are the odds in favor of getting at least one head
in three successive flips of a coin?
P(no heads) = 0.5^3 = 0.125
P(one or more heads) = 1 – 0.125 = 0.875
Odds in favor = p/(1-p) = 0.875/(1-0.875) = 7 (which is 7 to 1)
36. On a TV game show, the contestant is asked to select a
door and then is rewarded with the prize behind the door
selected. If the doors can be selected with equal probability,
what is the expected value of the selection if the
three doors have behind them a $40,000 foreign car, a $3
silly straw, and a $50 mathematics textbook?
40000/3 + 3/3 + 50/3
= $13351
38. Sam bought 1 of 250 tickets selling for $2 in a game
with a grand prize of $400. Was $2 a fair price to pay for
a ticket to play this game?
EV = 400/250 = $1.60
This was not a fair price, since the EV is lower than the ticket price.
48. The game of dots is played by rolling a fair die and
receiving $1 for each dot showing on the top face of the
die. What cost should be set for each roll if the game is
to be considered a fair game?
EV = 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6
= $3.50
2. In each case, consider what you know about the distribution
and then explain why you would expect it to be
or not to be normally distributed.
a. The wealth of the parents of students attending your
school
I would say yes, there is an average wealth, and then some people have lots more, but some don’t have nearly as much.
b. The values that a group of fourth-grade students
would give for the length of a segment that they
measured with a ruler
Yes, the mean will be approximately the true length, but some people will measure too long or too short.
c. The SAT or ACT examination scores in mathematics
for students who were in your high school
graduation class
Yes, there will be an average score, with some people doing better and some doing worse.
d. The weights of all incoming freshman students at
your school
Yes, there will be a mean weight, but some people will be overweight and some will be thin.
8. The number of accidents that occur at the intersection
of Pine and Linden streets between 3 p.m. and
6 p.m. on Friday afternoons is 0, 1, 2, or 3, with probabilities
of 0.84, 0.13, 0.02, and 0.01, respectively.
Graph this probability distribution. What is the
expected value for the random variable given the
number of accidents?
EV = 0*0.84 + 1*0.13 + 2*0.02 + 3*0.01
= 0.2 accidents
Graph:
12. The mean systolic blood pressure of adult males is normally
distributed with a mean of 138 (millimeters of
mercury) and a standard deviation of 9.7. What percent
of adult males have blood pressure between 161.28 and
164.9?
Z(161.28) = (161.28-138)/9.7 = 2.4
Z(164.9) = (164.9-138)/9.7 = 2.773
Prob(2.4 < z < 2.773) = 0.54%
14. A study of motor vehicle rates in the 50 states reveals
that traffic death rates (deaths per 100 million motor
vehicle miles driven) can be modeled by the normal
curve. The data suggest that the distribution has a mean
of 5.3 and a standard deviation of 1.3. Sketch the normal
curve, showing the mean and standard deviation.
20. Battery Power Problem. A certain type of thermal battery
for an airplane navigation device backup power has a
mean life of 300 hours with a standard deviation of 15
hours. What proportion of these batteries can be
expected to have lives of 322 hours or less? Assume a normal
distribution of backup power device lives.
Z(322) = (322-300)/15 = 1.466666
Prob(z < 1.4666666666) = 0.9288
2. How many 4-character license plates are possible with 2
letters from the alphabet followed by 2 digits, if repetitions
are allowed?
26*26*10*10
= 67600
if repetitions are not allowed?
26*25*10*9
= 58500
4. How many batting lineups of the nine players can be
made for a baseball team if the catcher bats first, the
shortstop second, and the pitcher last?
There are 6 other batters to place…
6*5*4*3*2*1
= 720
8. A class elects two officers, a president and a secretary/
treasurer, from its 12 members. How many different
ways can these two offices be filled from the members of
the class?
There are 12 ways to pick the pres. Then there are 11 people left for the sec/treas:
12*11
= 132
10. Five numbers are to be picked, without repetition, from 44
numbers to determine the winner of the Fortune Five
game in the state lottery. If the order of the numbers is
insignificant, how many different ways can a winning
quintuple be selected?
C(44,5)
= 44! / (5! * (44-5)!)
= 1086008
What is the probability of winning?
1/1086008
26. A ship carries exactly 10 different signal flags. If each
possible combination and ordering of 4 of these flags
connotes a specific message, how many signals can be
sent with these flags, taken 4 at a time?
There are 10 ways to pick the first flag, then 9, 8, and 7 for the last flag:
10*9*8*7
= 5040
28. A student asks, “What’s wrong with the argument that
the probability of rolling a double 6 in two rolls of a die
is 1/3 because 1/6+1/6=1/3 ” Write an explanation of your
understanding of the student’s misconception.
You cannot add the probabilities, you have to multiply them. If you did it this way, and you rolled 7 times, you would come out with a prob above 1, which doesn’t make sense at all!
38. Estimate the number of personally constructed greeting
cards possible at a machine if there are 12 designs, 30
messages, 18 closings, and 10 different paper stocks on
which to print the card. Indicate how you made your
estimate. How valid was your estimate?
Estimate: 10*30*20*10 = 60000
Exact: 12*30*18*10 = 64800
The estimate was a bit low.
2. A jar contains four marbles, each a different color: red,
blue, green, and yellow. If you draw two marbles from
the jar, one after another, replacing the first before
drawing the second, what is the probability of getting
- two red marbles?
¼*1/4 = 1/16
b. a red marble on the first draw and a green marble on
the second draw?
¼*1/4 = 1/16
- at least one red marble and one green marble?
1/16 + 1/16 = 1/8
- no yellow marbles?
(1-1/4)*(1-1/4)
¾*3/4 = 9/16
12. Suppose that pizzas can be ordered in four sizes (small,
medium, large, and Illini-size), with three crust choices
(thin, thick, and Chicago style), four choices of meat
(sausage, pepperoni, hamburger, and none) and two
types of cheese (regular or double). How many different
styles of pizza can be ordered?
Multiply:
4*3*4*2 = 96 choices