Probability Theory and Mathematical Systems Probability
Mathematical Systems
Probability
Solutions by Bracket
A First Course in Probability
Chapter 4—Problems
4. Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman (for instance, X = 1 if the top-ranked person is female). Find P X = i , i = 1, 2, 3, . . . , 8, 9, 10.
Let Ei be the event that the the ith scorer is female. Then the event X = i correspdonds to the cc event E1 E2 · · · Ei . It follows that cc P X = i = P (E1 E2 · · · Ei )
.
c c c c c
= P (E1 )P (E2 |E1 ) · · · P (Ei |E1 · · · Ei−1 )
Thus we have
P X=i i 1/
1
2
5/
2
18
5/
3
36
5/
4
84
5/
5
252
1/
6
252
0.
7, 8, 9, 10
12. In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither players guess correctly, then no money is exchanged. Consider a specified player and denote by X the amount of money he wins in a single game of Two-Finger Morra.
a. If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4 possibilities is equally likely, what are the possible values of X and what are their associated probabilities?
A given player can only win 0, ±2, ±3, or ±4 dollars. Consider two players A and B , and let X denote player A’s winnings. Let Aij denote the event that player A shows i fingers and guesses j , and define Bij similarly for player B.
1
We have P X = 2 = P (A11 B12 ) = P (A11 )P (B12 ) = 1 · 1 = 16 , since we have assumed that
44
1
Aij and Bij are independent and that P (Aij
Probability
Solutions by Bracket
A First Course in Probability
Chapter 4—Problems
4. Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman (for instance, X = 1 if the top-ranked person is female). Find P X = i , i = 1, 2, 3, . . . , 8, 9, 10.
Let Ei be the event that the the ith scorer is female. Then the event X = i correspdonds to the cc event E1 E2 · · · Ei . It follows that cc P X = i = P (E1 E2 · · · Ei )
.
c c c c c
= P (E1 )P (E2 |E1 ) · · · P (Ei |E1 · · · Ei−1 )
Thus we have
P X=i i 1/
1
2
5/
2
18
5/
3
36
5/
4
84
5/
5
252
1/
6
252
0.
7, 8, 9, 10
12. In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither players guess correctly, then no money is exchanged. Consider a specified player and denote by X the amount of money he wins in a single game of Two-Finger Morra.
a. If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4 possibilities is equally likely, what are the possible values of X and what are their associated probabilities?
A given player can only win 0, ±2, ±3, or ±4 dollars. Consider two players A and B , and let X denote player A’s winnings. Let Aij denote the event that player A shows i fingers and guesses j , and define Bij similarly for player B.
1
We have P X = 2 = P (A11 B12 ) = P (A11 )P (B12 ) = 1 · 1 = 16 , since we have assumed that
44
1
Aij and Bij are independent and that P (Aij