MATH 1333 Take home Exam Two ANSWERS
Problem 1 (70 points) We roll two fair six sided dice. Find the following probabilities:
(a) A sum of seven
(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)
(b) A sum of eight
(2,6),(3,5),(4,4),(5,3),(6,2)
(c) A sum of nine
(d) A sum of ten
3
36
0
36
(f) A sum of seven OR a sum of ten
i A five on the first die
4
36
(4,6),(5,5),(6,4)
=
1
6
1
9
1
12
=
=0
(1,6),(2,5),(3,4),(4,3),(5,2),(6,1),(4,6),(5,5),(6,4)
6
36
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
ii A five on the second die
=
5
36
(3,6),(4,5),(5,4),(6,3)
(e) A sum of seven AND a sum of ten
6
36
(5,6),(5,5),(5,4),(5,3),(5,2),(5,1)
=
1
6
6
36
=
9
36
=
1
4
1
6
(1,6),(2,5),(3,4),(4,3),(5,2),(6,1), (5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
iii A sum of seven OR a five on the first die
11
36
1
36
(5,2)
iv A sum of seven AND a five on the first die
Problem 2 (70 points) For each of the following situations, find BOTH the probability of and the odds for, or in favor
of the given event.
1
2
(a) Flip one coin and it comes up heads
1:1
3
4
(b) Flip two coins and we get at least one head
7
8
(c) Flip three coins and we get at least one head
1
6
(d) Roll one die and it comes up six
3:1
7:1
1:5
11
36
(e) Roll two dice and we get at least one six
91
216
(f) Roll three dice and we get at least one six
11 : 15
91 : 125
Problem 3 (70 points) Rob and Lydia are going to play a game. They roll one fair die. If an even number comes up,
Lydia gets $1. If an odd number comes up, Rob gets $1. Find the expected value of this game from Rob’s perspective in
each of the following cases:
(a) The die has six sides
( 63 (1) + 36 (−1) = 0
(b) The die has seven sides
( 73 (−1) + 74 (1) =
(c) The die has eight sides
( 48 (1) + 48 (−1) = 0
(d) The die has nine sides
(e) The die has ten sides
( 49 (−1) + 59 (1) =
5
( 10
(1) +
5
10 (−1)
1
7
1
9
=0
= 0.143
= 0.111
Problem 4 (70 points) Rob and Lydia are going to play a new game. They roll one fair die. If an even number comes
up, Lydia gets the amount showing in dollars. (So $4 if a four comes up) If an odd number comes up, Rob gets the amount
showing in dollars. (So $3 if a three comes up) Find the expected value of this game from Rob’s perspective in each of the
following cases:
1
6 (1)
(a) The die has six sides
+ 16 (−2) + 16 (3) + 16 (−4) + 61 (5) + 16 (−6) = − 21 = −0.50
(b) The die has seven sides
1
7 (1)
(c) The die has eight sides
1
8 (1)
1
9 (1)
(d) The die has nine sides
(e) The die has ten sides
− 12 = 0.50
+ 17 (−2) + 17 (3) + 17 (−4) + 17 (5) + 71 (−6) + 71 (7) =
+
1
8 (−2)
+
1
8 (3)
+
1
8 (−4)
+
1
8 (5)
+
1
8 (−6)
+
1
8 (7)
+
4
7
= 0.571
1
8 (−8)
= − 12 = 0.50
+ 19 (−2) + 91 (3) + 19 (−4) + 19 (5) + 19 (−6) + 19 (7) + 19 (−8) + 19 (9) =
5
9
= 0.556
1
1
1
1
1
1
1
1
1
1
10 (1) + 10 (−2) + 10 (3) + 10 (−4) + 10 (5) + 10 (−6) + 10 (7) + 10 (−8) + 10 (9) + 10 (−10)
Problem 5 (70 points) Use the following table of information to answer the following conditional questions.
Topping
Cheese
Meat
Veggie
Freshman
1
5
9
Sophmore
2
6
10
Junior
3
7
11
Senior
4
8
12
1. A students favorite topping is Veggie if we know they are a Junior
11
= 0.5238
3 + 7 + 11
2. A students favorite topping is Veggie if we know they are not a Junior
9 + 10 + 12
= 0.543
1 + 2 + 4 + 5 + 6 + 8 + 9 + 10 + 12
3. A students favorite topping is not Veggie if we know they are a Junior
3+7
= 0.1754
3 + 7 + 11
4. A students favorite topping is not Veggie if we know they are not a Junior
1+2+4+5+6+8
= 0.456
1 + 2 + 4 + 5 + 6 + 8 + 9 + 10 + 12
5. A students favorite topping is Veggie if we know they are a Junior or a Senior
11 + 12
= 0.511
3 + 4 + 7 + 8 + 11 + 12
6. A students favorite topping is Veggie if we know they are a Junior and a Senior
No one is both a Junior and a Senior so 0.
7. A students favorite topping is Veggie or Cheese if we know they are a Junior
3 + 11
= 0.667
3 + 7 + 11
8. A students favorite topping is Veggie or Cheese if we know they are not a Junior
1 + 2 + 4 + 9 + 10 + 12
= 0.667
1 + 2 + 4 + 5 + 6 + 8 + 9 + 10 + 12
9. A students favorite topping is Veggie or Cheese if we know they are a Junior or a Senior
3 + 4 + 11 + 12
= 0.667
3 + 4 + 7 + 8 + 11 + 12
10. A students favorite topping is Veggie or Cheese if we know they are not a Junior or not a Senior
1 + 2 + 3 + 4 + 9 + 10 + 11 + 12
= 0.667
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
11. A students favorite topping is Veggie or Cheese if we know they are not a Junior and not a Senior
1 + 2 + 9 + 10
= 0.667
=