Probability
Exercise 14.1 Solutions
Key Definitions:
Sample Space ($S$): The set of all possible outcomes.
Event ($E$): Any subset of $S$.
Mutually Exclusive: Events $A$ and $B$ are mutually exclusive if $A \cap B = \phi$.
Exhaustive Events: Events whose union is the sample space $S$.
Sample Space ($S$): The set of all possible outcomes.
Event ($E$): Any subset of $S$.
Mutually Exclusive: Events $A$ and $B$ are mutually exclusive if $A \cap B = \phi$.
Exhaustive Events: Events whose union is the sample space $S$.
Q1
1. Die rolled. E: “die shows 4”, F: “die shows even number”. Are they mutually exclusive?
Sample Space
$S = \{1, 2, 3, 4, 5, 6\}$
Events
$E = \{4\}$
$F = \{2, 4, 6\}$
$F = \{2, 4, 6\}$
Intersection
$E \cap F = \{4\} \cap \{2, 4, 6\} = \{4\}$
Conclusion
Since $E \cap F \neq \phi$, they are not mutually exclusive.
No
Q2
2. Die thrown. Describe A (x<7), B (x>7), C (mult 3), D (x<4), E (even>4), F (not<3).
Define Sets
$S = \{1, 2, 3, 4, 5, 6\}$
$A = \{1, 2, 3, 4, 5, 6\}$ (x < 7)
$B = \phi$ (x > 7 is impossible)
$C = \{3, 6\}$ (multiples of 3)
$D = \{1, 2, 3\}$ (x < 4)
$E = \{6\}$ (even > 4)
$F = \{3, 4, 5, 6\}$ (not less than 3)
$A = \{1, 2, 3, 4, 5, 6\}$ (x < 7)
$B = \phi$ (x > 7 is impossible)
$C = \{3, 6\}$ (multiples of 3)
$D = \{1, 2, 3\}$ (x < 4)
$E = \{6\}$ (even > 4)
$F = \{3, 4, 5, 6\}$ (not less than 3)
Operations
$A \cup B = A \cup \phi = A = \{1, 2, 3, 4, 5, 6\}$
$A \cap B = A \cap \phi = \phi$
$B \cup C = \phi \cup \{3, 6\} = \{3, 6\}$
$E \cap F = \{6\} \cap \{3, 4, 5, 6\} = \{6\}$
$D \cap E = \{1, 2, 3\} \cap \{6\} = \phi$
$A – C = \{1, 2, 3, 4, 5, 6\} – \{3, 6\} = \{1, 2, 4, 5\}$
$D – E = \{1, 2, 3\} – \{6\} = \{1, 2, 3\}$
$A \cap B = A \cap \phi = \phi$
$B \cup C = \phi \cup \{3, 6\} = \{3, 6\}$
$E \cap F = \{6\} \cap \{3, 4, 5, 6\} = \{6\}$
$D \cap E = \{1, 2, 3\} \cap \{6\} = \phi$
$A – C = \{1, 2, 3, 4, 5, 6\} – \{3, 6\} = \{1, 2, 4, 5\}$
$D – E = \{1, 2, 3\} – \{6\} = \{1, 2, 3\}$
Complements
$F’ = S – F = \{1, 2\}$
$E \cap F’ = \{6\} \cap \{1, 2\} = \phi$
$E \cap F’ = \{6\} \cap \{1, 2\} = \phi$
Q3
3. Rolling a pair of dice. A: sum > 8, B: 2 on either die, C: sum $\ge$ 7 and multiple of 3.
Set A
Sum 9, 10, 11, 12: $\{(3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)\}$
Set B
2 on either: $\{(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (1,2), (3,2), (4,2), (5,2), (6,2)\}$
Set C
Sum $\ge$ 7 & mult of 3 (Sum 9 or 12): $\{(3,6), (4,5), (5,4), (6,3), (6,6)\}$
Check Pairs
$A \cap B$: Do any in A have a 2? No. $A \cap B = \phi$. (Mutually Exclusive)
$B \cap C$: Do any in C have a 2? No. $B \cap C = \phi$. (Mutually Exclusive)
$A \cap C$: Common elements are sum 9 and 12. $\{(3,6), (4,5), (5,4), (6,3), (6,6)\} \neq \phi$.
$B \cap C$: Do any in C have a 2? No. $B \cap C = \phi$. (Mutually Exclusive)
$A \cap C$: Common elements are sum 9 and 12. $\{(3,6), (4,5), (5,4), (6,3), (6,6)\} \neq \phi$.
Pairs (A, B) and (B, C) are mutually exclusive.
Q4
4. Three coins tossed. A: 3 Heads, B: 2 Heads 1 Tail, C: 3 Tails, D: Head on 1st coin.
Sample Space
$S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$
Events
$A = \{HHH\}$
$B = \{HHT, HTH, THH\}$
$C = \{TTT\}$
$D = \{HHH, HHT, HTH, HTT\}$
$B = \{HHT, HTH, THH\}$
$C = \{TTT\}$
$D = \{HHH, HHT, HTH, HTT\}$
(i) Mut. Excl.
$A \cap B = \phi$, $A \cap C = \phi$, $B \cap C = \phi$.
$C \cap D = \phi$ (D has H, C has no H).
$A \cap D = \{HHH\} \neq \phi$. $B \cap D \neq \phi$.
Mutually exclusive pairs: $(A, B), (A, C), (B, C), (C, D)$.
$C \cap D = \phi$ (D has H, C has no H).
$A \cap D = \{HHH\} \neq \phi$. $B \cap D \neq \phi$.
Mutually exclusive pairs: $(A, B), (A, C), (B, C), (C, D)$.
(ii) Simple
Events with 1 element: A and C.
(iii) Compound
Events with >1 element: B and D.
Q5
5. Three coins tossed. Describe events matching specific criteria.
(i)
Two Mutually Exclusive: Getting 3 heads ($A=\{HHH\}$) and Getting 3 tails ($B=\{TTT\}$).
(ii)
3 Mutually Excl. & Exhaustive: A: 0 heads, B: Exactly 1 head, C: At least 2 heads.
(iii)
2 Not Mutually Excl: A: Getting at least 2 heads, B: Getting at least 1 head. (Intersection exists).
(iv)
2 Mutually Excl but NOT Exhaustive: A: 3 heads, B: 3 tails. Union is not $S$.
(v)
3 Mutually Excl but NOT Exhaustive: A: 3 heads, B: 3 tails, C: Exactly 1 head. (Missing ‘Exactly 2 heads’).
Q6
6. Two Dice. A: Even on 1st, B: Odd on 1st, C: Sum $\le$ 5. Describe events.
Sets
A = $\{(2,y), (4,y), (6,y) \mid y \in 1..6\}$ (18 elements)
B = $\{(1,y), (3,y), (5,y) \mid y \in 1..6\}$ (18 elements)
C = $\{(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)\}$
B = $\{(1,y), (3,y), (5,y) \mid y \in 1..6\}$ (18 elements)
C = $\{(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)\}$
(i) A’
Not even on 1st = Odd on 1st = B.
(ii) not B
Not odd on 1st = Even on 1st = A.
(iii) A or B
Even on 1st $\cup$ Odd on 1st = All outcomes = $S$.
(iv) A and B
Number on 1st die cannot be even AND odd. $\phi$.
(v) A but not C
$A – C$: Elements in A with Sum > 5.
A starts with 2,4,6. Remove C’s: (2,1), (2,2), (2,3), (4,1).
Remaining A: $\{(2,4)..(2,6), (4,2)..(4,6), (6,1)..(6,6)\}$.
A starts with 2,4,6. Remove C’s: (2,1), (2,2), (2,3), (4,1).
Remaining A: $\{(2,4)..(2,6), (4,2)..(4,6), (6,1)..(6,6)\}$.
(vi) B or C
$B \cup C$: Odd on 1st OR Sum $\le 5$. Join set B with $\{(2,1), (2,2), (2,3), (4,1)\}$.
(vii) B and C
Odd on 1st AND Sum $\le 5$: $\{(1,1), (1,2), (1,3), (1,4), (3,1), (3,2)\}$.
(viii)
$A \cap B’ \cap C’ = A \cap A \cap C’ = A \cap C’ = A – C$. (Same as part v).
Q7
7. True/False based on Q6.
(i)
A and B are mutually exclusive. True (Even vs Odd on 1st die).
(ii)
A and B are mutually exclusive and exhaustive. True ($A \cap B = \phi$ and $A \cup B = S$).
(iii)
$A = B’$. True.
(iv)
A and C are mutually exclusive. False. Intersection exists e.g., (2,1) is in A and C.
(v)
A and B’ are mutually exclusive. False. $B’ = A$. $A \cap A = A \neq \phi$.
(vi)
A’, B’, C are mutually exclusive and exhaustive. False. $A’=B, B’=A$. Union is $S$ (exhaustive), but C overlaps with both A and B, so not mutually exclusive.