# 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