Section 10.1
(1) First we complete the table by calculating the missing value…in this case
since we know that the probabilities must add up to ONE, so the missing
value must equal 1-(0.1+0.4+0.3) = 0.2…giving the following table:
Outcome
Value
Probability
Red
$2
0.1
Yellow
Blue
-$5
$1
0.2
0.4
Cost to play: $1
Green
$3
0.3
Next, calculate the gross value of the “game” by multiplying the value by the
probability for each, and then adding together (as follows):
2(0.1)+(-5)(0.2)+1(0.4)+3(0.3) = 0.5
And finally subtract the cost to play from the gross value:
0.5 – 1 = -$0.50
(2) First we complete the table by calculating the missing value…in this case
since we know that the probabilities must add up to ONE, so the missing
value must equal 1-(4/52+12/52) = 36/52…giving the following table:
Outcome
Value
Probability
Ace
$10
4/52
Face Card
-$10
12/52
Cost to play: $1
Other
$3
36/52
Next, calculate the gross value of the “game” by multiplying the value by the
probability for each, and then adding together (as follows):
10(4/52)+(-10)(12/52)+3(36/52) = 28/52
And finally subtract the cost to play from the gross value:
28/52 – 1 = -$24/52 or -$0.46
(3) This is the same as problem #1 except the last step, which is:
0.5 – 2 = -$1.50
(4) This is the same as problem #2 except the last step, which is:
28/52 – 2 = -$76/52 or -$1.46
(5) The definition of a fair game is when the net value of the game is ZERO; or
another way to say it is that the gross value of the game equals the cost per
play of the game. So since in problem #1 the gross value was $0.5, then if
the cost per play equals that it is fair…so the answer is $0.50
(6) The definition of a fair game is when the net value of the game is ZERO; or
another way to say it is that the gross value of the game equals the cost per
play of the game. So, in this case we will need to “adjust” the value of
other to equal the cost per play ($1). So if we replace the $3 under “other”
in problem #2 and do the calculation for the gross value, you get:
10(4/52)+(-10)(12/52)+x(36/52) = -80/52+36x/52
Now setting this equal to one and solve for x…
-80/52+36x/52 = 1
(multiply both sides by 52)
-80+36x = 52
(add 80 to both sides)
36x =132
(divide both sides by 36)
x = 132/36 = $3.67
(7) Using the table format from problems 1 & 2, let’s put the data from this
problem into the table…
Outcome
Value
Probability
First
$1000
Second
$500
Third
$100
Cost to play: $1
Loss
$0
Now the probabilities are a bit harder, but if you consider that the total
number of tickets sold is 10,000 and that there is one ticket for each winner,
then for First prize the probability is 1/10000, and so forth, giving the following
table:
Outcome
Value
Probability
First
$1000
1/10000
Second
Third
$500
$100
2/10000
5/10000
Cost to play: $1
Loss
$0
9992/10000
So then the gross value is:
1000(1/10000)+500(2/10000)+100(5/10000)+0(9992/10000) = 0.25
and the net value is: 0.25 – 1 = -$0.75
(8) Using the table format from problems 1 & 2, let’s put the data from this
problem into the table…
Outcome
Value
Probability
Double
$5
Differ by 3+
-$2
Other
$1
Cost to play: $0.50
Now the probabilities are a bit harder, but using the table on page 35, you can
see that there are 6 places that the numbers match (doubles)…so 6/36. For
the ones that differ by 3 or more, let’s first look at the green numbers…if the
green is ONE then the red must be 4, 5 or 6; if the green is TWO then the red
must be 5 or 6; and if the green is THREE then the red must be 6; also if you
turned it around for the RED you would get the same results, so there are 12
ways to answer this question, 12/36…giving the following table:
Outcome
Value
Probability
Double
$5
6/36
Differ by 3+
-$2
12/36
Cost to play: $0.50
Other
$1
18/36
So then the gross value is:
5(6/36)+(-2)(12/36)+1(18/36) = 24/36
and the net value is: 24/36 – 0.5 = $6/36 or $0.17
The definition of a fair game is when the net value of the game is ZERO; or
another way to say it is that the gross value of the game equals the cost per
play of the game. So since the gross value is 24/36, then if the cost per play
equals that it is fair…so the answer is $24/36 or $0.67
(9) Using the table format from problems 1 & 2, let’s put the data from this
problem into the table…
Outcome
Value
Probability
Same Suit
$5
Same Denom
$2
Other
$0
Cost to play: $0.50
Now for the probabilities we will use combinations. (Remember that a
probability like this will be the number of ways that “answer the question”
divided by the total number of ways possible.) So the Total Number of Ways
Possible is the number of ways of choosing 2 cards from a deck of 52 or
C(52,2)= 1326. Now the number of ways to get two cards from the same suit:
first it can be any of the four suits so you must first choose a suit [C(4,1)] and
then choose 2 cards from the 13 cards that make up that suit [C(13,2)]; which
gives you C(4,1)*C(13,2) = 312. So the probability is 312/1326 = 0.2353. Next
the same denomination will be done similarly, but in this case we do not know
which denomination (2,3,4,5,6,7,8,9,10,J,Q,K,A) so we must choose one of the
13 denominations [C(13,1)] and then from the four cards in that denomination
you will choose 2 of those cards [C(4,2)], giving you: C(13,1)*C(4,2) = 78. So
the probability is 78/1326 = 0.0588. And for other it will be whatever is left
over so that the total probability will equal ONE.
Outcome
Value
Probability
Same Suit
$5
0.2353
Same Denom
$2
0.0588
Cost to play: $0.50
Other
$0
0.7059
So then the gross value is:
5(0.2353)+( 2)(0.0588)+0(0.7059) = 1.2943
and the net value is: 1.2943 – 0.5 = $0.79
The definition of a fair game is when the net value of the game is ZERO; or
another way to say it is that the gross value of the game equals the cost per
play of the game. So since the gross value is 1.2943, then if the cost per
play equals that it is fair…so the answer is $1.29
(10)
Using the table format from problems 1 & 2, let’s put the data from
this problem into the table…
Outcome
Value
Probability
Win
$10000
Lose
$0
Cost to play: $0.30
Now the probabilities are basic, one winner in 2 Million or a probability of
1/2000000, and to lose is the remainder (1999999/2000000)…
Outcome
Value
Probability
Win
Lose
$10000
$0
1/2000000 1999999/2000000
Cost to play: $0.30
So then the gross value is:
10000(1/2000000)+0(1999999/2000000) = 0.005
and the net value is: 0.005 – 0.3 = 0.295 or $0.30
(11)
This can be answered any way you wish, there is no set answer. But
one such answer could be that people enter because there is a chance and
it doesn’t cost that much time.
(12)
What people are NOT seeing in the game of problem #10 is that the
people make their money because people believe that there is a “better”
chance if you BUY something (which is not true), so the actual cost of them
playing is more than the 30 cents (it is also the cost of the magazine, or
whatever).
(13)
Using the table format from problems 1 & 2, let’s put the data from
this problem into the table…
Outcome
Value
Probability
Lose Camera Retain Camera
$10000
$0
0.20
0.80
Cost to play: $1200
So then the gross value is:
10000(0.20)+0(0.80) = 2000
and the net value is: 2000 – 1200 = $800
This net of $800 says that it is a good investment, if you were to do this many
times; but in reality (with an 80% probability of NOT losing your equipment)
and you are probably only doing it this one time, it is highly likely that you will
just “lose” $1200, so you must decide to risk it or count that as part of the cost
of making the documentary.
The definition of a fair game is when the net value of the game is ZERO; or
another way to say it is that the gross value of the game equals the cost per
play of the game. So since the gross value is 2000, then if the cost per play
equals that it is fair…so the answer is $2000