\displaystyle \textbf{Question 1: }\text{Cards each marked with one of the numbers }4,5,6,\ldots,20\text{ are placed in a}
\displaystyle \text{box and mixed thoroughly. One card is drawn at random from the box. What is the probability}
\displaystyle \text{of getting an even number?}
\displaystyle \text{Answer:}
\displaystyle \text{Total number of cards}=20-4+1=17.
\displaystyle \text{Even numbers are }4,6,8,10,12,14,16,18,20.
\displaystyle \therefore P(\text{an even number})=\frac{9}{17}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{One card is drawn from a well shuffled deck of }52\text{ playing cards. What is the}
\displaystyle \text{probability of getting a non-face card?}
\displaystyle \text{Answer:}
\displaystyle \text{Number of face cards}=12.
\displaystyle \therefore \text{Number of non-face cards}=52-12=40.
\displaystyle \therefore P(\text{a non-face card})=\frac{40}{52}=\frac{10}{13}.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{A bag contains }5\text{ red, }8\text{ green and }7\text{ white balls. One ball is drawn at}
\displaystyle \text{random from the bag. What is the probability of getting a white ball or a green ball?}
\displaystyle \text{Answer:}
\displaystyle \text{Total number of balls}=5+8+7=20.
\displaystyle \text{Number of white or green balls}=7+8=15.
\displaystyle \therefore P(\text{a white or a green ball})=\frac{15}{20}=\frac{3}{4}.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{A die is thrown once. What is the probability of getting a prime number?}
\displaystyle \text{Answer:}
\displaystyle \text{Prime numbers on a die are }2,3,5.
\displaystyle \therefore P(\text{a prime number})=\frac{3}{6}=\frac{1}{2}.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{A die is thrown once. What is the probability of getting a number lying}
\displaystyle \text{between }2\text{ and }6\text{?}
\displaystyle \text{Answer:}
\displaystyle \text{Numbers lying between }2\text{ and }6\text{ are }3,4,5.
\displaystyle \therefore P(\text{a number lying between }2\text{ and }6)=\frac{3}{6}=\frac{1}{2}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{A die is thrown once. What is the probability of getting an odd number?}
\displaystyle \text{Answer:}
\displaystyle \text{Odd numbers on a die are }1,3,5.
\displaystyle \therefore P(\text{an odd number})=\frac{3}{6}=\frac{1}{2}.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{If }\bar{E}\text{ denote the complement or negation of an event }E,\text{ what is the value}
\displaystyle \text{of }P(E)+P(\bar{E})\text{?}
\displaystyle \text{Answer:}
\displaystyle \text{For an event }E\text{ and its complement }\bar{E},
\displaystyle P(\bar{E})=1-P(E).
\displaystyle \therefore P(E)+P(\bar{E})=1.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{One card is drawn at random from a well shuffled deck of }52\text{ cards. What is the}
\displaystyle \text{probability of getting an ace?}
\displaystyle \text{Answer:}
\displaystyle \text{Number of aces in a deck of }52\text{ cards}=4.
\displaystyle \therefore P(\text{an ace})=\frac{4}{52}=\frac{1}{13}.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Two coins are tossed simultaneously. What is the probability of getting at least}
\displaystyle \text{one head?}
\displaystyle \text{Answer:}
\displaystyle \text{The possible outcomes are }HH,HT,TH,TT.
\displaystyle \text{Favourable outcomes are }HH,HT,TH.
\displaystyle \therefore P(\text{at least one head})=\frac{3}{4}.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Tickets numbered }1\text{ to }20\text{ are mixed up and then a ticket is drawn at random.}
\displaystyle \text{What is the probability that the ticket drawn bears a number which is a multiple of }3\text{?}
\displaystyle \text{Answer:}
\displaystyle \text{Multiples of }3\text{ from }1\text{ to }20\text{ are }3,6,9,12,15,18.
\displaystyle \therefore \text{Number of favourable outcomes}=6.
\displaystyle \therefore P(\text{a multiple of }3)=\frac{6}{20}=\frac{3}{10}.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{From a well shuffled pack of cards, a card is drawn at random. Find the}
\displaystyle \text{probability of getting a black queen.}\hfill\text{[CBSE 2008]}
\displaystyle \text{Answer:}
\displaystyle \text{There are }2\text{ black queens in a pack of }52\text{ cards.}
\displaystyle \therefore P(\text{a black queen})=\frac{2}{52}=\frac{1}{26}.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{A die is thrown once. Find the probability of getting a number less than }3.
\displaystyle \hfill\text{[CBSE 2008, 20]}
\displaystyle \text{Answer:}
\displaystyle \text{Numbers less than }3\text{ are }1,2.
\displaystyle \therefore P(\text{a number less than }3)=\frac{2}{6}=\frac{1}{3}.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{Two coins are tossed simultaneously. Find the probability of getting exactly}
\displaystyle \text{one head.}\hfill\text{[CBSE 2009]}
\displaystyle \text{Answer:}
\displaystyle \text{The possible outcomes are }HH,HT,TH,TT.
\displaystyle \text{Favourable outcomes are }HT,TH.
\displaystyle \therefore P(\text{exactly one head})=\frac{2}{4}=\frac{1}{2}.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{A die is thrown once. What is the probability of getting a number greater}
\displaystyle \text{than }4\text{?}\hfill\text{[CBSE 2010]}
\displaystyle \text{Answer:}
\displaystyle \text{Numbers greater than }4\text{ are }5,6.
\displaystyle \therefore P(\text{a number greater than }4)=\frac{2}{6}=\frac{1}{3}.
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{What is the probability that a number selected at random from the numbers}
\displaystyle 3,4,5,\ldots,9\text{ is a multiple of }4\text{?}\hfill\text{[CBSE 2010]}
\displaystyle \text{Answer:}
\displaystyle \text{The numbers are }3,4,5,6,7,8,9.\text{ Thus, total number of outcomes}=7.
\displaystyle \text{Multiples of }4\text{ are }4,8.
\displaystyle \therefore P(\text{a multiple of }4)=\frac{2}{7}.
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{A letter of English alphabet is chosen at random. Determine the probability}
\displaystyle \text{that the chosen letter is a consonant.}\hfill\text{[CBSE 2015, 20]}
\displaystyle \text{Answer:}
\displaystyle \text{There are }26\text{ letters in the English alphabet, of which }5\text{ are vowels.}
\displaystyle \therefore \text{Number of consonants}=26-5=21.
\displaystyle \therefore P(\text{a consonant})=\frac{21}{26}.
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{A bag contains }3\text{ red and }5\text{ black balls. A ball is drawn at random from the}
\displaystyle \text{bag. What is the probability that the ball drawn is not red?}\hfill\text{[CBSE 2017]}
\displaystyle \text{Answer:}
\displaystyle \text{Total number of balls}=3+5=8.
\displaystyle \text{Number of balls which are not red}=5.
\displaystyle \therefore P(\text{not red})=\frac{5}{8}.
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{A number is chosen at random from the numbers }-3,-2,-1,0,1,2,3.\text{ What will}
\displaystyle \text{be the probability that the square of this number is less than or equal to }1\text{?}
\displaystyle \hfill\text{[CBSE 2017]}
\displaystyle \text{Answer:}
\displaystyle \text{Total number of outcomes}=7.
\displaystyle \text{The numbers whose squares are less than or equal to }1\text{ are }-1,0,1.
\displaystyle \therefore P(\text{square of the number}\leq1)=\frac{3}{7}.
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{If the probability of winning a game is }0.07,\text{ what is the probability of losing it?}
\displaystyle \hfill\text{[CBSE 2020]}
\displaystyle \text{Answer:}
\displaystyle P(\text{losing})=1-P(\text{winning}).
\displaystyle =1-0.07=0.93.
\displaystyle \therefore P(\text{losing})=0.93.
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{If a number is chosen at random from the numbers }-3,-2,-1,0,1,2,3,
\displaystyle \text{what is the probability that }x^2<4\text{?}\hfill\text{[CBSE 2020]}
\displaystyle \text{Answer:}
\displaystyle \text{Total number of outcomes}=7.
\displaystyle \text{The numbers satisfying }x^2<4\text{ are }-1,0,1.
\displaystyle \therefore P(x^2<4)=\frac{3}{7}.
\displaystyle \\

\displaystyle \textbf{Question 21: }\text{A bag contains }4\text{ red, }3\text{ blue and }2\text{ yellow balls. One ball is drawn at}
\displaystyle \text{random from the bag. Find the probability that drawn ball is (i) red (ii) yellow.}
\displaystyle \hfill\text{[CBSE 2023]}
\displaystyle \text{Answer:}
\displaystyle \text{Total number of balls}=4+3+2=9.
\displaystyle \text{(i) }P(\text{red ball})=\frac{4}{9}.

\displaystyle \text{(ii) }P(\text{yellow ball})=\frac{2}{9}.
\displaystyle \\

\displaystyle \textbf{Question 22: }\text{A fair coin is tossed twice, find the probability of getting `at most one}
\displaystyle \text{head'.}\hfill\text{[CBSE 2023]}
\displaystyle \text{Answer:}
\displaystyle \text{The possible outcomes are }HH,HT,TH,TT.
\displaystyle \text{Favourable outcomes for at most one head are }HT,TH,TT.
\displaystyle \therefore P(\text{at most one head})=\frac{3}{4}.
\displaystyle \\

\displaystyle \textbf{Question 23: }\text{The probability of guessing the correct answer of a certain test question is}
\displaystyle \frac{x}{12}.\text{ If the probability of not guessing the correct answer is }\frac{5}{6},\text{ then find the value of }x.
\displaystyle \hfill\text{[CBSE 2025]}
\displaystyle \text{Answer:}
\displaystyle P(\text{correct answer})+P(\text{not correct answer})=1.
\displaystyle \therefore \frac{x}{12}+\frac{5}{6}=1.
\displaystyle \frac{x}{12}=1-\frac{5}{6}=\frac{1}{6}.
\displaystyle \therefore x=2.
\displaystyle \\

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