\displaystyle \textbf{Question 1: }\text{A coin is tossed. Find the total number of elementary}
\displaystyle \text{events and also the total number of events associated with the}
\displaystyle \text{random experiment.}
\displaystyle \textbf{Answer:}
\displaystyle \text{When a coin is tossed, the sample space is }S=\{H,T\}.
\displaystyle \text{Hence, the number of elementary events}=2.
\displaystyle \text{The number of events equals the number of subsets of }S=2^2=4.
\displaystyle \therefore \text{Total elementary events}=2\text{ and total events}=4.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{List all events associated with the random experiment of}
\displaystyle \text{tossing two coins. How many of them are elementary events?}
\displaystyle \textbf{Answer:}
\displaystyle \text{The sample space is }S=\{HH,HT,TH,TT\}.
\displaystyle \text{The events are all subsets of }S:
\displaystyle \phi,\{HH\},\{HT\},\{TH\},\{TT\},\{HH,HT\},\{HH,TH\},
\displaystyle \{HH,TT\},\{HT,TH\},\{HT,TT\},\{TH,TT\},
\displaystyle \{HH,HT,TH\},\{HH,HT,TT\},\{HH,TH,TT\},
\displaystyle \{HT,TH,TT\},\{HH,HT,TH,TT\}.
\displaystyle \text{Hence, the total number of events}=2^4=16.
\displaystyle \text{The elementary events are }\{HH\},\{HT\},\{TH\}\text{ and }\{TT\}.
\displaystyle \therefore \text{The number of elementary events is }4.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Three coins are tossed once. Describe the following events}
\displaystyle \text{associated with this random experiment: }A=\text{Getting three heads,}
\displaystyle B=\text{Getting two heads and one tail, }C=\text{Getting three tails,}
\displaystyle D=\text{Getting a head on the first coin.}
\displaystyle \textbf{(i) }\text{Which pairs of events are mutually exclusive?}
\displaystyle \textbf{(ii) }\text{Which events are elementary events?}
\displaystyle \textbf{(iii) }\text{Which events are compound events?}
\displaystyle \textbf{Answer:}
\displaystyle S=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}.
\displaystyle A=\{HHH\},\qquad B=\{HHT,HTH,THH\}.
\displaystyle C=\{TTT\},\qquad D=\{HHH,HHT,HTH,HTT\}.
\displaystyle A\cap B=\varnothing,\qquad A\cap C=\varnothing,\qquad A\cap D=\{HHH\}.
\displaystyle B\cap C=\varnothing,\qquad B\cap D=\{HHT,HTH\},\qquad C\cap D=\varnothing.
\displaystyle \text{Events having empty intersection are mutually exclusive.}
\displaystyle \text{(i) }\therefore (A,B),(A,C),(B,C)\text{ and }(C,D)\text{ are mutually exclusive pairs.}
\displaystyle \text{(ii) }A\text{ and }C\text{ each contain one sample point. Hence, they are elementary events.}
\displaystyle \text{(iii) }B\text{ and }D\text{ each contain more than one sample point. Hence, they are compound events.}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{In a single throw of a die, describe the following events:}
\displaystyle \textbf{(i) }A\text{ - Getting a number less than }7\qquad\textbf{(ii) }B\text{ - Getting a number greater than }7
\displaystyle \textbf{(iii) }C\text{ - Getting a multiple of }3\qquad\textbf{(iv) }D\text{ - Getting a number less than }4
\displaystyle \textbf{(v) }E\text{ - Getting an even number greater than }4\qquad\textbf{(vi) }F\text{ - Getting a number not less than }3
\displaystyle \text{Also find }A\cup B,\ A\cap B,\ B\cap C,\ E\cap F,\ D\cap F\text{ and }\overline{F}.
\displaystyle \textbf{Answer:}
\displaystyle S=\{1,2,3,4,5,6\}.
\displaystyle A=\{1,2,3,4,5,6\}.
\displaystyle B=\varnothing.
\displaystyle C=\{3,6\}.
\displaystyle D=\{1,2,3\}.
\displaystyle E=\{6\}.
\displaystyle F=\{3,4,5,6\}.
\displaystyle A\cup B=\{1,2,3,4,5,6\}.
\displaystyle A\cap B=\varnothing.
\displaystyle B\cap C=\varnothing.
\displaystyle E\cap F=\{6\}.
\displaystyle D\cap F=\{3\}.
\displaystyle \overline{F}=S-F=\{1,2\}.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Three coins are tossed. Describe:}
\displaystyle \text{(i) two events }A\text{ and }B\text{ which are mutually exclusive.}
\displaystyle \text{(ii) three events }A,B\text{ and }C\text{ which are mutually exclusive and exhaustive.}
\displaystyle \text{(iii) two events }A\text{ and }B\text{ which are not mutually exclusive.}
\displaystyle \text{(iv) two events }A\text{ and }B\text{ which are mutually exclusive but not exhaustive.}
\displaystyle \textbf{Answer:}
\displaystyle S=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}.
\displaystyle \text{(i) Let }A\text{ be the event of getting no Tail and }B\text{ be the event}
\displaystyle \text{of getting no Head.}
\displaystyle A=\{HHH\},\qquad B=\{TTT\}.
\displaystyle A\cap B=\varnothing.
\displaystyle \therefore A\text{ and }B\text{ are mutually exclusive.}
\displaystyle \text{(ii) Let }A\text{ be the event of getting no Head, }B\text{ be the event}
\displaystyle \text{of getting exactly one Head and }C\text{ be the event of getting}
\displaystyle \text{at least two Heads.}
\displaystyle A=\{TTT\}.
\displaystyle B=\{HTT,THT,TTH\}.
\displaystyle C=\{HHT,HTH,THH,HHH\}.
\displaystyle A\cap B=\varnothing,\qquad B\cap C=\varnothing,\qquad C\cap A=\varnothing.
\displaystyle A\cup B\cup C=S.
\displaystyle \therefore A,B\text{ and }C\text{ are mutually exclusive and exhaustive.}
\displaystyle \text{(iii) Let }A\text{ be the event of getting three Heads and }B\text{ be the event}
\displaystyle \text{of getting at least two Heads.}
\displaystyle A=\{HHH\},\qquad B=\{HHH,HHT,HTH,THH\}.
\displaystyle A\cap B=\{HHH\}\neq\varnothing.
\displaystyle \therefore A\text{ and }B\text{ are not mutually exclusive.}
\displaystyle \text{(iv) Let }A\text{ be the event of getting exactly one Head and }B\text{ be}
\displaystyle \text{the event of getting exactly one Tail.}
\displaystyle A=\{HTT,THT,TTH\}.
\displaystyle B=\{HHT,HTH,THH\}.
\displaystyle A\cap B=\varnothing\text{ but }A\cup B\neq S.
\displaystyle \therefore A\text{ and }B\text{ are mutually exclusive but not exhaustive.}
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{A die is thrown twice. Each time, the number appearing on it}
\displaystyle \text{is recorded. Describe the following events:}
\displaystyle \text{(i) }A=\text{Both numbers are odd.}
\displaystyle \text{(ii) }B=\text{Both numbers are even.}
\displaystyle \text{(iii) }C=\text{The sum of the numbers is less than }6.
\displaystyle \text{Also find }A\cup B,\ A\cap B,\ A\cup C\text{ and }A\cap C.
\displaystyle \text{Which pairs of events are mutually exclusive?}
\displaystyle \textbf{Answer:}
\displaystyle \text{When a die is thrown twice, the total number of outcomes}=6^2=36.
\displaystyle A=\{(1,1),(1,3),(1,5),(3,1),(3,3),(3,5),
\displaystyle \phantom{A=\{}(5,1),(5,3),(5,5)\}.
\displaystyle B=\{(2,2),(2,4),(2,6),(4,2),(4,4),(4,6),
\displaystyle \phantom{B=\{}(6,2),(6,4),(6,6)\}.
\displaystyle C=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),
\displaystyle \phantom{C=\{}(2,3),(3,1),(3,2),(4,1)\}.
\displaystyle A\cup B=\{(1,1),(1,3),(1,5),(3,1),(3,3),(3,5),
\displaystyle \phantom{A\cup B=\{}(5,1),(5,3),(5,5),(2,2),(2,4),(2,6),
\displaystyle \phantom{A\cup B=\{}(4,2),(4,4),(4,6),(6,2),(6,4),(6,6)\}.
\displaystyle A\cap B=\varnothing.
\displaystyle A\cup C=\{(1,1),(1,2),(1,3),(1,4),(1,5),(2,1),
\displaystyle \phantom{A\cup C=\{}(2,2),(2,3),(3,1),(3,2),(3,3),(3,5),
\displaystyle \phantom{A\cup C=\{}(4,1),(5,1),(5,3),(5,5)\}.
\displaystyle A\cap C=\{(1,1),(1,3),(3,1)\}.
\displaystyle \text{Since }A\cap B=\varnothing,\text{ the events }A\text{ and }B\text{ are mutually exclusive.}
\displaystyle \text{Also, }A\cap C\neq\varnothing\text{ and }B\cap C=\{(2,2)\}\neq\varnothing.
\displaystyle \therefore \text{Only }A\text{ and }B\text{ are mutually exclusive.}
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Two dice are thrown. The events }A,B,C,D,E\text{ and }F
\displaystyle \text{are described as follows:}
\displaystyle A=\text{Getting an even number on the first die.}
\displaystyle B=\text{Getting an odd number on the first die.}
\displaystyle C=\text{Getting at most }5\text{ as the sum of the numbers on the two dice.}
\displaystyle D=\text{Getting a sum greater than }5\text{ but less than }10.
\displaystyle E=\text{Getting at least }10\text{ as the sum of the numbers on the dice.}
\displaystyle F=\text{Getting an odd number on exactly one of the dice.}
\displaystyle \text{(i) Describe }A\text{ and }B,\ B\text{ or }C,\ B\text{ and }C,\ A\text{ and }E,
\displaystyle A\text{ or }F,\text{ and }A\text{ and }F.
\displaystyle \text{(ii) State whether the given statements are true or false.}
\displaystyle \textbf{Answer:}
\displaystyle \text{When two dice are thrown, the total number of outcomes}=6^2=36.
\displaystyle A=\{(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),
\displaystyle \phantom{A=\{}(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),
\displaystyle \phantom{A=\{}(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}.
\displaystyle B=\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
\displaystyle \phantom{B=\{}(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),
\displaystyle \phantom{B=\{}(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\}.
\displaystyle C=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),
\displaystyle \phantom{C=\{}(2,3),(3,1),(3,2),(4,1)\}.
\displaystyle D=\{(1,5),(1,6),(2,4),(2,5),(2,6),(3,3),(3,4),
\displaystyle \phantom{D=\{}(3,5),(3,6),(4,2),(4,3),(4,4),(4,5),
\displaystyle \phantom{D=\{}(5,1),(5,2),(5,3),(5,4),(6,1),(6,2),(6,3)\}.
\displaystyle E=\{(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)\}.
\displaystyle F=\{(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),
\displaystyle \phantom{F=\{}(3,2),(3,4),(3,6),(4,1),(4,3),(4,5),
\displaystyle \phantom{F=\{}(5,2),(5,4),(5,6),(6,1),(6,3),(6,5)\}.
\displaystyle \text{(i) }A\text{ and }B=A\cap B=\varnothing.
\displaystyle B\text{ or }C=B\cup C
\displaystyle =\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
\displaystyle \phantom{=\{}(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),
\displaystyle \phantom{=\{}(3,4),(3,5),(3,6),(4,1),(5,1),(5,2),
\displaystyle \phantom{=\{}(5,3),(5,4),(5,5),(5,6)\}.
\displaystyle B\text{ and }C=B\cap C
\displaystyle =\{(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)\}.
\displaystyle A\text{ and }E=A\cap E
\displaystyle =\{(4,6),(6,4),(6,5),(6,6)\}.
\displaystyle A\text{ or }F=A\cup F
\displaystyle =\{(1,2),(1,4),(1,6),(2,1),(2,2),(2,3),(2,4),
\displaystyle \phantom{=\{}(2,5),(2,6),(3,2),(3,4),(3,6),(4,1),(4,2),
\displaystyle \phantom{=\{}(4,3),(4,4),(4,5),(4,6),(5,2),(5,4),(5,6),
\displaystyle \phantom{=\{}(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}.
\displaystyle A\text{ and }F=A\cap F
\displaystyle =\{(2,1),(2,3),(2,5),(4,1),(4,3),(4,5),
\displaystyle \phantom{=\{}(6,1),(6,3),(6,5)\}.
\displaystyle \text{(ii)(a) True, since }A\cap B=\varnothing.
\displaystyle \text{(b) True, since }A\cap B=\varnothing\text{ and }A\cup B=S.
\displaystyle \text{(c) False, since }A\cap C=\{(2,1),(2,2),(2,3),(4,1)\}\neq\varnothing.
\displaystyle \text{(d) False, since }C\cap D=\varnothing\text{ but }C\cup D\neq S.
\displaystyle \text{(e) True, since }C\cap D=D\cap E=E\cap C=\varnothing
\displaystyle \text{and }C\cup D\cup E=S.
\displaystyle \text{(f) True, since }A'=B,\ B'=A\text{ and }A'\cap B'=\varnothing.
\displaystyle \text{(g) False. Although }A\cup B\cup F=S,\text{ the events are not}
\displaystyle \text{mutually exclusive because }A\cap F\neq\varnothing\text{ and }B\cap F\neq\varnothing.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{The numbers }1,2,3\text{ and }4\text{ are written separately on four}
\displaystyle \text{slips of paper. The slips are put in a box and mixed thoroughly.}
\displaystyle \text{Two slips are drawn one after the other, without replacement.}
\displaystyle \text{Describe the following events:}
\displaystyle A=\text{The number on the first slip is larger than that on the second slip.}
\displaystyle B=\text{The number on the second slip is greater than }2.
\displaystyle C=\text{The sum of the numbers on the two slips is }6\text{ or }7.
\displaystyle D=\text{The number on the second slip is twice that on the first slip.}
\displaystyle \text{Which pair or pairs of events are mutually exclusive?}
\displaystyle \textbf{Answer:}
\displaystyle \text{Since two slips are drawn without replacement, the sample space is}
\displaystyle S=\{(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),
\displaystyle \phantom{S=\{}(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)\}.
\displaystyle A=\{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)\}.
\displaystyle B=\{(1,3),(1,4),(2,3),(2,4),(3,4),(4,3)\}.
\displaystyle C=\{(2,4),(3,4),(4,2),(4,3)\}.
\displaystyle D=\{(1,2),(2,4)\}.
\displaystyle A\cap B=\{(4,3)\}\neq\varnothing.
\displaystyle A\cap C=\{(4,2),(4,3)\}\neq\varnothing.
\displaystyle A\cap D=\varnothing.
\displaystyle B\cap C=\{(2,4),(3,4),(4,3)\}\neq\varnothing.
\displaystyle B\cap D=\{(2,4)\}\neq\varnothing.
\displaystyle C\cap D=\{(2,4)\}\neq\varnothing.
\displaystyle \therefore A\text{ and }D\text{ are the only mutually exclusive events.}
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{A card is picked up from a deck of }52\text{ playing cards.}
\displaystyle \textbf{(i) }\text{What is the sample space of the experiment?}
\displaystyle \textbf{(ii) }\text{What is the event that the chosen card is a black face card?}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) The sample space consists of all }52\text{ cards of a standard deck.}
\displaystyle \text{Hence, }S=\{\text{all }52\text{ playing cards}\}.
\displaystyle \text{(ii) Let }E\text{ be the event that the chosen card is a black face card.}
\displaystyle E=\{\text{Jack of Clubs},\text{Queen of Clubs},\text{King of Clubs},
\displaystyle \phantom{E=\{}\text{Jack of Spades},\text{Queen of Spades},\text{King of Spades}\}.
\displaystyle \\

 

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