ECE316 Additional Tutorial for the week of June 1-5
Problems from Textbook – 8th Edition – Chapter 4 – Page 172
Problem 4.1:
Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that
we win $2 for each black ball selected and we lose $1 for each white ball selected. Let X denote our
winnings. What are the possible values of X, and what are the probabilities associated with each value?
Solution:
•
•
•
Assume that selecting the orange ball will not affect winning or losing.
Selecting 2 balls out of 8+4+2 = 14 total balls, we will have a total number of combinations of
14
ቀ ቁ
2
The possible values of X are :
Balls selected
Value of X
Number of combinations
4
2 black
4
ቀ ቁ
2
4
2
1 black & 1 orange
2
ቀ ቁ×ቀ ቁ
1
1
4
8
1 black & 1 white
1
ቀ ቁ×ቀ ቁ
1
1
2
2 orange
0
ቀ ቁ
2
8
2
1 orange & 1 white
-1
ቀ ቁ×ቀ ቁ
1
1
8
2 white
-2
ቀ ቁ
2
The associated probabilities of different values of X
ቀ4ቁ
6
ܲ ሺܺ = 4ሻ = 2 =
,
ቀ14ቁ 91
2
4
ቀ ቁ × ቀ8ቁ 32
1 =
,
ܲ ሺܺ = 1ሻ = 1
14
91
ቀ ቁ
2
8
ቀ ቁ × ቀ2ቁ 16
1 =
ܲ ሺܺ = −1ሻ = 1
,
14
91
ቀ ቁ
2
ቀ4ቁ × ቀ2ቁ
1 = 8
ܲ ሺܺ = 2ሻ = 1
91
ቀ14ቁ
2
ቀ2ቁ
1
ܲ ሺܺ = 0ሻ = 2 =
ቀ14ቁ 91
2
ቀ8ቁ
28
ܲ ሺܺ = −2ሻ = 2 =
ቀ14ቁ 91
2
Problem 4.2:
Two fair dice are rolled. Let X equal the product of the 2 dice, compute ܲሺܺ = ݅ ሻ for ݅ = 1,2, … ,36
Solution:
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
4
6
8
10
12
3
3
6
9
12
15
18
4
4
8
12
16
20
24
5
5
10
15
20
25
30
6
6
12
18
24
30
36
The corresponding probabilities will be
Value of X
1
Number of
occurrences
1
2
2
3
2
4
3
5
2
6
4
7
0
8
2
9
1
10
2
11
0
12
4
13
0
14
0
15
2
16
1
17
0
18
2
19
0
20
2
21
0
22
0
Associated
probability
1ൗ
36
2ൗ
36
2ൗ
36
3ൗ
36
2ൗ
36
4ൗ
36
0ൗ
36
2ൗ
36
1ൗ
36
2ൗ
36
0ൗ
36
4ൗ
36
0ൗ
36
0ൗ
36
2ൗ
36
1ൗ
36
0ൗ
36
2ൗ
36
0ൗ
36
2ൗ
36
0ൗ
36
0ൗ
36
Value of X
23
Number of
occurrences
0
24
2
25
1
26
0
27
0
28
0
29
0
30
2
31
0
32
0
33
0
34
0
35
0
36
1
Associated
probability
0ൗ
36
2ൗ
36
1ൗ
36
0ൗ
36
0ൗ
36
0ൗ
36
0ൗ
36
2ൗ
36
0ൗ
36
0ൗ
36
0ൗ
36
0ൗ
36
0ൗ
36
1ൗ
36
Problem 4.7:
Suppose that a die is rolled twice. What are the possible values of the following random variables can
take on:
(a)
(b)
(c)
(d)
The maximum value to appear in the two rolls;
The minimum value to appear in the two rolls;
The sum of the two rolls;
The value of the first roll minus the value of the second roll?
Solution:
(a) Maximum value to appear in the two rolls X = max (roll1, roll2)
ܺ = ሼ1,2,3,4,5,6ሽ
(b) Minimum value to appear in the two rolls X = min (roll1, roll2)
ܺ = ሼ1,2,3,4,5,6ሽ
(c) Sum of the 2 rolls X = sum(roll1, roll2)
ܺ = ሼ2,3,4,5,6,7,8,9,10,11,12ሽ
st
(d) Value of the 1 roll minus the value of the second roll X = roll1 – roll2
ܺ = ሼ−5, −4, −3, −2, −1,0,1,2,3,4,5ሽ
Problem 4.8:
If the die in problem 7 is assumed fair, calculate the probabilities associated with the random variables
in part (a) through (d)
Solution:
Since the die is assumed to be fair, in any roll the probability of having any number will be 1ൗ6 , so any
pair of combinations will have a probability of 1ൗ36
(a) Maximum value to appear in the two rolls X = max (roll1, roll2)
ଵ
ଵ
ଵ
X = 1 only one combination (1,1) ܲ ሺܺ = 1ሻ = ଺ × ଺ = ଷ଺
ଵ
ଵ
X = 2 3 combinations (2,1), (1,2) , (2,2) ܲ ሺܺ = 2ሻ = 3 × × =
଺
଺
ହ
X = 3 5 combinations (3,1),(3,2),(1,3),(2,3),(3,3) ܲሺܺ = 3ሻ = ଷ଺
ଷ
ଷ଺
଻
X = 4 7 combinations (4,1),(4,2),(4,3),(1,4),(2,4),(3,4),(4,4) ܲ ሺܺ = 4ሻ = ଷ଺
ଽ
X = 5 9 combinations (5,1), ..... ,(5,4),(1,5),(2,5),.....,(4,5),(5,5) ܲ ሺܺ = 5ሻ = ଷ଺
X = 6 11 combinations (6,1),.....,(6,5),(1,6),(2,6),.....,(5,6),(6,6) ܲሺܺ = 6ሻ =
ଵଵ
ଷ଺
(b) Minimum value to appear in the two rolls X = min (roll1, roll2)
ଵ
ଵ
ଵଵ
X = 1 (1,2),(1,3),.....(1,6),(6,1),(5,1),....,(2,1),(1,1) ܲሺܺ = 1ሻ = 11 × ଺ × ଺ = ଷ଺
ଵ
ଵ
ଽ
X = 2 (2,3), (2,4),...(2,6),(6,2),(5,2),....(3,2), (2,2) ܲሺܺ = 2ሻ = 9 × ଺ × =
଺
ଷ଺
଻
X = 3 (3,4),(3,5),(3,6),(6,3),(5,3),(4,3)(3,3) ܲሺܺ = 3ሻ = ଷ଺
ହ
X = 4 (4,5),(4,6),(6,4),(5,4),(4,4) ܲ ሺܺ = 4ሻ = ଷ଺
ଷ
X = 5 (5,6),(6,5),(5,5) ܲሺܺ = 5ሻ = ଷ଺
ଵ
X = 6 (6,6) ܲሺܺ = 6ሻ = ଷ଺
(c) Sum of the 2 rolls X= sum(roll1, roll2)
ଵ
ଵ
ଵ
X = 2 (1,1) ܲ ሺܺ = 2ሻ = ଺ × ଺ = ଷ଺
X = 3 (1,2),(2,1) ܲሺܺ = 3ሻ =
ଶ
ଷ଺
ଷ
X = 4 (1,3),(3,1),(2,2) ܲሺܺ = 4ሻ = ଷ଺
ସ
X = 5 (1,4),(4,1),(2,3),(3,2) ܲሺܺ = 5ሻ = ଷ଺
ହ
X = 6 (1,5),(5,1),(2,4),(4,2),(3,3) ܲ ሺܺ = 6ሻ = ଷ଺
X = 7 (1,6),(6,1),(2,5),(5,2),(3,4),(4,3) ܲሺܺ = 7ሻ =
ହ
X = 8 (2,6),(6,2),(3,5),(5,3),(4,4) ܲ ሺܺ = 8ሻ = ଷ଺
଺
ଷ଺
ସ
X = 9 (3,6),(6,3),(4,5),(5,4) ܲሺܺ = 9ሻ = ଷ଺
ଷ
X = 10 (4,6),(6,4),(5,5) ܲ ሺܺ = 10ሻ = ଷ଺
ଶ
X = 11 (5,6),(6,6) ܲሺܺ = 11ሻ = ଷ଺
X = 12 (6,6) ܲሺܺ = 12ሻ =
ଵ
ଷ଺
(d) Value of the 1st roll minus the value of the second roll X = roll1 – roll2
ଵ
X = -5 (1,6) ܲ ሺܺ = −5ሻ = ଷ଺
X = -4 (1,5),(2,6) ܲሺܺ = −4ሻ =
ଶ
ଷ଺
ଷ
X = -3 (1,4),(2,5),(3,6) ܲሺܺ = −3ሻ = ଷ଺
X = -2 (1,3),(2,4),(3,5),(4,6) ܲ ሺܺ = −2ሻ =
ସ
ଷ଺
ହ
X = -1 (1,2),(2,3),(3,4),(4,5),(5,6) ܲ ሺܺ = −1ሻ = ଷ଺
଺
X = 0 (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) ܲሺܺ = 0ሻ = ଷ଺
ହ
X = 1 (6,5),(5,4),(4,3),(3,2),(2,1) ܲ ሺܺ = 1ሻ = ଷ଺
ସ
X = 2 (6,4),(5,3),(4,2),(3,1) ܲሺܺ = 2ሻ = ଷ଺
ଷ
X = 3 (6,3),(5,2),(4,1) ܲሺܺ = 3ሻ = ଷ଺
ଶ
X = 4 (6,2),(5,1) ܲሺܺ = 4ሻ = ଷ଺
ଵ
X = 5 (6,1) ܲሺܺ = 5ሻ = ଷ଺
Problem 4.22:
Suppose that 2 teams play a series of games that ends when one of them has won ݅ games. Suppose
that each game played is, independently, won by team A with probability p. Find the expected number
of games that are played when
(a) ݅ = 2
(b) ݅ = 3
Also, show in both cases that this number is maximized when ‫ = ݌‬1ൗ2
Solution:
Let N be the number of games played.
(a) ݅ = 2
A wins 2 games out of 2 ‫݌‬ଶ
B wins 2 games out of 2 ሺ1 − ‫݌‬ሻଶ
3
A wins 2 games out of 3 ‫݌‬ଶ × ሺ1 − ‫݌‬ሻ × ቄቀ ቁ − 1ቅ
2
3
ଶ
B wins 2 games out of 3 ሺ1 − ‫݌‬ሻ × ‫ × ݌‬ቄቀ ቁ − 1ቅ
2
3
3
∴ ‫ܧ‬ሾܰሿ = 2 × ሺ‫݌‬ଶ + ሺ1 − ‫݌‬ሻଶ ሻ + 3 × ቀ‫݌‬ଶ × ሺ1 − ‫݌‬ሻ × ቄቀ ቁ − 1ቅ + ሺ1 − ‫݌‬ሻଶ × ‫ × ݌‬ቄቀ ቁ − 1ቅቁ
2
2
ଶ
= −2‫ ݌‬+ 2‫ ݌‬+ 2
To find the maximum number of games,
ௗ
ௗ௣
ଵ
‫ܧ‬ሾܰሿ = −4‫ ݌‬+ 2 = 0 ‫ = ݌‬ଶ
(b) ݅ = 3
A wins 3 games out of 3 ‫݌‬ଷ
B wins 3 games out of 3 ሺ1 − ‫݌‬ሻଷ
4
A wins 3 games out of 4 ‫݌‬ଷ × ሺ1 − ‫݌‬ሻ × ቄቀ ቁ − 1ቅ
3
4
ଷ
B wins 3 games out of 4 ሺ1 − ‫݌‬ሻ × ‫ × ݌‬ቄቀ ቁ − 1ቅ
3
5
A wins 3 games out of 5 ‫݌‬ଷ × ሺ1 − ‫݌‬ሻଶ × ቄቀ ቁ − 4ቅ
3
5
B wins 3 games out of 5 ሺ1 − ‫݌‬ሻଷ × ‫݌‬ଶ × ቄቀ ቁ − 4ቅ
3
4
4
∴ ‫ܧ‬ሾܰሿ = 3 × ሺ‫݌‬ଷ + ሺ1 − ‫݌‬ሻଷ ሻ + 4 × ቀ‫݌‬ଷ × ሺ1 − ‫݌‬ሻ × ቄቀ ቁ − 1ቅ + ሺ1 − ‫݌‬ሻଷ × ‫ × ݌‬ቄቀ ቁ − 1ቅቁ +
3
3
5
5
ଷ
ଶ
ଷ
ଶ
ସ
ଷ
ଶ
5 × ቀ‫ × ݌‬ሺ1 − ‫݌‬ሻ × ቄቀ ቁ − 4ቅ + ሺ1 − ‫݌‬ሻ × ‫ × ݌‬ቄቀ ቁ − 4ቅቁ = 6‫ ݌‬− 12‫ ݌‬+ 3‫ ݌‬+ 3‫ ݌‬+ 3
3
3
To find the maximum number of games,
ௗ
ௗ௣
ଵ
‫ܧ‬ሾܰሿ = 24‫݌‬ଷ − 36‫݌‬ଶ + 6‫ ݌‬+ 3 = 0 ‫ = ݌‬ଶ , ±1.207
ଵ
Then ‫ = ݌‬, and we will refuse the other 2 values because one of them is negative and the other is
ଶ
greater than 1.
Problem 4.25:
Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with
probability 0.7. Assume that the results of the flips are independent, and let X equal the total number of
heads that result.
(a) Find P{X=1}
(b) Determine E[X]
Solution:
This experiment is performed one time only.
X = number of heads
X
0
1
2
P(X)
ሺ1 − 0.6ሻ × ሺ1 − 0.7ሻ = 0.12
ሺ0.6ሻ × ሺ1 − 0.7ሻ + ሺ1 − 0.6ሻ × ሺ0.7ሻ = 0.46
ሺ0.6ሻ × ሺ0.7ሻ = .42
(a) ܲ ሺܺ = 1ሻ = 0.46
(b) ‫ܧ‬ሾܺሿ = 0 × ܲ ሺܺ = 0ሻ + 1 × ܲሺܺ = 1ሻ + 2 × ܲ ሺܺ = 2ሻ = 0.46 + 0.84 = 1.3