Chapter 5 – Probability
Format: Questions in Red • Answers in Green • Mobile-friendly MathJax.
Topics: Random Experiment • Outcomes • Equally likely • Sample Space & Events • Probability \(P(E)=\dfrac{n(E)}{n(S)}\) • Cards • Dice • Coins • Counting.
🚩 20 Most-Important 1-Mark Questions & Solutions
Q1. Define a random experiment.
Ans. An experiment with known possible outcomes, but each trial’s result is uncertain and outcomes are equally likely (e.g., toss, die).
Q2. What is an outcome?
Ans. A single possible result of a random experiment (e.g., \(H\) on a coin toss).
Q3. Write the sample space when a die is thrown once.
Ans. \(S=\{1,2,3,4,5,6\}\), \(n(S)=6\).
Q4. Two coins are tossed. How many outcomes?
Ans. \(S=\{HH,HT,TH,TT\}\), so \(n(S)=4\).
Q5. Define an event.
Ans. A set of favourable outcomes; a subset of the sample space.
Q6. State the probability formula.
Ans. \(P(A)=\dfrac{n(A)}{n(S)}\), \(0\le P(A)\le 1\).
Q7. Probability of getting head on a fair coin?
Ans. \(\dfrac{1}{2}\).
Q8. Probability of getting 1 on a fair die?
Ans. \(\dfrac{1}{6}\).
Q9. How many red cards in a pack?
Ans. \(26\) (13 hearts + 13 diamonds).
Q10. How many face cards in a pack?
Ans. \(12\) (K,Q,J in each of 4 suits).
Q11. If \(n(S)=50\) and \(n(A)=8\), find \(P(A)\).
Ans. \(P(A)=\dfrac{8}{50}=\dfrac{4}{25}\).
Q12. Sample space for two dice?
Ans. Ordered pairs \((i,j)\) with \(i,j\in\{1,\dots,6\}\); \(n(S)=36\).
Q13. Define equally likely outcomes.
Ans. Outcomes with same chance of occurring (e.g., faces of a fair die).
Q14. What is a sure event?
Ans. An event with probability \(1\).
Q15. What is an impossible event?
Ans. An event with probability \(0\).
Q16. If \(P(E)=0.25\), express as percentage.
Ans. \(25\%\).
Q17. In cards, probability of drawing an ace?
Ans. \(\dfrac{4}{52}=\dfrac{1}{13}\).
Q18. Probability of getting a red card?
Ans. \(\dfrac{26}{52}=\dfrac{1}{2}\).
Q19. If \(P(E)=\dfrac{3}{5}\), then \(P(\overline{E})=\ ?\)
Ans. \(1-\dfrac{3}{5}=\dfrac{2}{5}\).
Q20. Which is more likely: head on coin or ‘1’ on die?
Ans. Head on coin (\(\tfrac{1}{2}>\tfrac{1}{6}\)).
📝 20 Most-Important 2-Mark Questions & Solutions
Q1. One coin and one die are thrown. Write \(S\) and find \(P(\text{H and even})\).
Ans. \(S=\{(H,1)...(H,6),(T,1)...(T,6)\}\), \(n(S)=12\). Favourable: \((H,2),(H,4),(H,6)\Rightarrow 3\). So \(P=\dfrac{3}{12}=\dfrac{1}{4}\).
Q2. Two coins tossed. Find \(P(\text{at least one head})\).
Ans. \(S=\{HH,HT,TH,TT\}\). Favourable: \(HH,HT,TH\Rightarrow 3\). \(P=\dfrac{3}{4}\).
Q3. One die tossed. Find \(P(\text{prime})\) and \(P(\text{even})\).
Ans. Prime set \(\{2,3,5\}\Rightarrow \dfrac{3}{6}=\dfrac{1}{2}\). Even set \(\{2,4,6\}\Rightarrow \dfrac{1}{2}\).
Q4. From numbers \(1\)–\(50\), find \(P(\text{multiple of }6)\).
Ans. Multiples: \(6,12,18,24,30,36,42,48\Rightarrow 8\). So \(P=\dfrac{8}{50}=\dfrac{4}{25}\).
Q5. Pack of cards: \(P(\text{face card})\).
Ans. Face cards \(=12\). \(P=\dfrac{12}{52}=\dfrac{3}{13}\).
Q6. Two dice rolled. Find \(P(\text{sum is }7)\).
Ans. Pairs: \((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\Rightarrow 6\). \(P=\dfrac{6}{36}=\dfrac{1}{6}\).
Q7. Two dice rolled. Find \(P(\text{same numbers})\).
Ans. Doubles: \((1,1)...(6,6)\Rightarrow 6\). \(P=\dfrac{6}{36}=\dfrac{1}{6}\).
Q8. A bag has \(5\) strawberry, \(6\) coffee, \(2\) peppermint. Find \(P(\text{coffee})\) and \(P(\text{peppermint})\).
Ans. Total \(=13\). \(P(\text{coffee})=\dfrac{6}{13}\), \(P(\text{peppermint})=\dfrac{2}{13}\).
Q9. Two-digit numbers using \(2,3,5\) without repetition. Write \(S\) and \(n(S)\).
Ans. \(S=\{23,25,32,35,52,53\}\), \(n(S)=6\).
Q10. If \(P(A)=\dfrac{1}{5}\) and \(n(A)=2\), find \(n(S)\).
Ans. \(\dfrac{2}{n(S)}=\dfrac{1}{5}\Rightarrow n(S)=10\).
Q11. A card from \(1\)–\(25\). Find \(P(\text{perfect square})\).
Ans. Squares: \(1,4,9,16,25\Rightarrow 5\). \(P=\dfrac{5}{25}=\dfrac{1}{5}\).
Q12. Two dice: \(P(\text{first}>\text{second})\).
Ans. \(15\) ordered pairs ⇒ \(P=\dfrac{15}{36}=\dfrac{5}{12}\).
Q13. Three coins: \(P(\text{no head})\) and \(P(\text{at least two heads})\).
Ans. \(P(\text{no head})=\dfrac{1}{8}\). “At least two” : \(HHH,HHT,HTH,THH\Rightarrow 4/8=\dfrac{1}{2}\).
Q14. Two dice: \(P(\text{sum multiple of }5)\).
Ans. Sums \(5,10\): counts \(4+3=7\Rightarrow \dfrac{7}{36}\).
Q15. From 52 cards, \(P(\text{spade or heart})\).
Ans. \(13+13=26\Rightarrow \dfrac{26}{52}=\dfrac{1}{2}\).
Q16. A spinner \(1\)–\(8\). Find \(P(\text{odd})\).
Ans. \(\{1,3,5,7\}\Rightarrow \dfrac{4}{8}=\dfrac{1}{2}\).
Q17. A bag: \(3\) white, \(4\) black, \(3\) red. \(P(\text{not black})\).
Ans. Total \(10\). Not black \(=6\Rightarrow \dfrac{6}{10}=\dfrac{3}{5}\).
Q18. Cards with letters of “MATHEMATICS” (11 cards). \(P(M)\).
Ans. \(m\) occurs \(2\Rightarrow \dfrac{2}{11}\).
Q19. Two dice: \(P(\text{sum at least }10)\).
Ans. Sums \(10,11,12\Rightarrow 3+2+1=6\Rightarrow \dfrac{1}{6}\).
Q20. Die once: \(P(\text{number}<3)\).
Ans. \(\{1,2\}\Rightarrow \dfrac{2}{6}=\dfrac{1}{3}\).
🎯 20 Most-Important 3-Mark Questions & Solutions
Q1. Two dice are rolled. Find \(P(\text{sum is prime})\).
Ans. Prime sums \(=2,3,5,7,11\). Counts \(1,2,4,6,2\Rightarrow 15\). \(P=\dfrac{15}{36}=\dfrac{5}{12}\).
Q2. Two dice: \(P(\text{sum is multiple of }6)\) and \(P(\text{sum}\ge 10)\).
Ans. Multiples of 6: \(6,12\Rightarrow 5+1=6\Rightarrow \dfrac{6}{36}=\dfrac{1}{6}\). \(\ge 10\Rightarrow \dfrac{6}{36}=\dfrac{1}{6}\).
Q3. From \(1\)–\(50\): \(P(\text{divisible by }6)\) and \(P(\text{perfect square})\).
Ans. \(\dfrac{8}{50}=\dfrac{4}{25}\) and squares \(=7\Rightarrow \dfrac{7}{50}\).
Q4. Cards: \(P(\text{red})\), \(P(\text{face})\), \(P(\text{red face})\).
Ans. \(\dfrac{26}{52}=\dfrac{1}{2}\), \(\dfrac{12}{52}=\dfrac{3}{13}\), red face \(=6\Rightarrow \dfrac{6}{52}=\dfrac{3}{26}\).
Q5. Three coins tossed. Find \(P(\text{exactly one head})\) and \(P(\text{at least one tail})\).
Ans. Exactly one head: \(HTT,THT,TTH\Rightarrow 3/8\). At least one tail \(=1-P(\text{HHH})=1-\dfrac{1}{8}=\dfrac{7}{8}\).
Q6. A committee of two from boys \(B_1,B_2,B_3\) and girls \(G_1,G_2\). Find \(P(\text{both boys}), P(\text{at least one girl})\).
Ans. Total \(\binom{5}{2}=10\). Both boys \(\binom{3}{2}=3\Rightarrow \dfrac{3}{10}\). At least one girl \(=1-\dfrac{3}{10}=\dfrac{7}{10}\).
Q7. Two-digit numbers using digits \(0,1,2,3,4\) (repetition allowed). Find \(P(\text{multiple of }4)\).
Ans. Total \(=4\times 5=20\) (tens \(1\)–\(4\)). Multiples of \(4\): \(12,20,24,32,40,44\Rightarrow 6\). \(P=\dfrac{6}{20}=\dfrac{3}{10}\).
Q8. Same setting as Q7: \(P(\text{prime})\) and \(P(\text{multiple of }11)\).
Ans. Primes: \(11,13,23,31,41,43\Rightarrow 6\Rightarrow \dfrac{3}{10}\). Multiples of \(11\): \(11,22,33,44\Rightarrow 4\Rightarrow \dfrac{1}{5}\).
Q9. Bag: \(3\) white, \(4\) black, \(3\) red. Find \(P(\text{white or red})\) and \(P(\text{not white})\).
Ans. \(P(\text{W or R})=\dfrac{6}{10}=\dfrac{3}{5}\). \(P(\text{not W})=\dfrac{7}{10}\).
Q10. Two dice: \(P(\text{first}>\text{second})\), \(P(\text{first}<\text{second})\), \(P(\text{equal})\).
Ans. \(\dfrac{15}{36},\ \dfrac{15}{36},\ \dfrac{6}{36}\) respectively.
Q11. A spinner \(1\)–\(8\). Find \(P(8)\), \(P(\text{odd})\), \(P(>2)\), \(P(<9)\).
Ans. \(\dfrac{1}{8},\ \dfrac{1}{2},\ \dfrac{3}{4},\ 1\).
Q12. Letters of “MATHEMATICS”. Find \(P(\text{vowel})\) and \(P(M)\).
Ans. Vowels \(=A,A,E,I\Rightarrow 4/11\). \(P(M)=2/11\).
Q13. If \(P(E)=0.6\) and total trials \(=200\), expected favourable outcomes?
Ans. \(0.6\times 200=120\).
Q14. A bag has \(4\) white, \(5\) black, \(1\) red. Find \(P(\text{black})\) and \(P(\text{non-black})\).
Ans. Total \(10\). Black \(=\dfrac{5}{10}=\dfrac{1}{2}\). Non-black \(=\dfrac{5}{10}=\dfrac{1}{2}\).
Q15. Two dice: \(P(\text{sum }=4\text{ or }8)\).
Ans. Sum 4: \(3\); sum 8: \(5\). Total \(8\Rightarrow \dfrac{8}{36}=\dfrac{2}{9}\).
Q16. Draw from 52 cards. \(P(\text{black or ace})\).
Ans. Black \(=26\), aces \(=4\), overlap (black aces) \(=2\). \(P=\dfrac{26+4-2}{52}=\dfrac{28}{52}=\dfrac{7}{13}\).
Q17. Three coins: \(P(\text{exactly two heads})\).
Ans. \(HHT,HTH,THH\Rightarrow 3/8\).
Q18. One die: \(P(\text{number not divisible by }3)\).
Ans. Divisible by 3: \(3,6\Rightarrow 2\). Complement: \(4\Rightarrow \dfrac{4}{6}=\dfrac{2}{3}\).
Q19. From \(1\)–\(15\): \(P(\text{even})\), \(P(\text{multiple of }5)\).
Ans. Even \(=7/15\). Multiple of \(5\) \(=3/15=1/5\).
Q20. A bag: \(3\) red, \(3\) white, \(3\) green. \(P(\text{red})\), \(P(\text{not red})\), \(P(\text{red or white})\).
Ans. \(\dfrac{1}{3},\ \dfrac{2}{3},\ \dfrac{2}{3}\) respectively.
📘 Textbook Exercise Questions & Perfect Solutions
Practice Set 5.1 — Possibilities
Q1(1). How many possibilities? Vanita picks one place from {Ajintha, Mahabaleshwar, Lonar Sarovar, Tadoba, Amboli, Raigad, Matheran, Anandvan}.
Ans. \(8\) possibilities.
Q1(2). Any day of a week chosen randomly.
Ans. \(7\) possibilities.
Q1(3). Select a card from \(52\) cards.
Ans. \(52\) possibilities.
Q1(4). Cards numbered \(10\) to \(20\) (one number per card). Select one.
Ans. \(11\) possibilities (inclusive \(10,11,\dots,20\)).
Let’s think. Which has more possibility: getting \(1\) on a die or getting head on a coin?
Ans. Head on a coin, since \(\tfrac{1}{2}>\tfrac{1}{6}\).
Practice Set 5.2 — Sample Space
Q1(1). One coin and one die thrown together: write \(S\) and \(n(S)\).
Ans. \(S=\{(H,1),(H,2),\dots,(H,6),(T,1),\dots,(T,6)\}\), \(n(S)=12\).
Q1(2). Two-digit numbers using digits \(2,3,5\) (no repetition): write \(S\) and \(n(S)\).
Ans. \(S=\{23,25,32,35,52,53\}\), \(n(S)=6\).
Q2. Spinner stops on a colour: {Red, Purple, Green, Orange, Yellow, Blue}. On which colour may it stop?
Ans. Any of the \(6\) colours (all equally likely).
Q3. In March 2019, list dates that are multiples of \(5\).
Ans. \(5,10,15,20,25,30\).
Q4. Form a Road Safety Committee of two from \(B_1,B_2,G_1,G_2\). Write all possibilities.
Ans. Boys-only: \(\{B_1B_2\}\); Girls-only: \(\{G_1G_2\}\); One boy–one girl: \(\{B_1G_1,B_1G_2,B_2G_1,B_2G_2\}\). Thus \(S=\{B_1B_2,G_1G_2,B_1G_1,B_1G_2,B_2G_1,B_2G_2\}\), \(n(S)=6\).
Practice Set 5.3 — Events
Q1(1). One die: write \(S\), and events A(even), B(odd), C(prime) with sizes.
Ans. \(S=\{1,2,3,4,5,6\}\). \(A=\{2,4,6\},\ n(A)=3;\ B=\{1,3,5\},\ n(B)=3;\ C=\{2,3,5\},\ n(C)=3.\)
Q1(2). Two dice: A(sum multiple of \(6\)), B(sum \(\ge 10\)), C(same digits).
Ans. \(n(S)=36\). \(n(A)=6\) (sums \(6,12\)); \(n(B)=6\) (sums \(10,11,12\)); \(n(C)=6\) (doubles).
Q1(3). Three coins: A(at least two heads), B(no head), C(head on 2nd coin).
Ans. \(S\) has 8 outcomes. \(A=\{HHT,HTH,THH,HHH\}, n=4\). \(B=\{TTT\}, n=1\). \(C=\{H H, T H\) with 2nd coin \(H\)\(\}=\{HHH,HHT,THH,THT\}, n=4\).
Q1(4). Two-digit numbers from \(0,1,2,3,4,5\) (no repetition). A(even), B(divisible by 3), C(>50).
Ans. Total \(=5\times 5=25\). A: units even \(\Rightarrow\) count \(= (tens\ 1\text{–}5)\times (units\ 0,2,4)\) minus repetition with tens if even; enumerating gives \(15\). B: list multiples of 3; count \(=8\). C: tens \(=5\) units \(=0\text{–}4\Rightarrow 5\). (Any equivalent clear listing earns full credit.)
Q1(5). Environment committee of two from men \(M_1,M_2,M_3\) and women \(W_1,W_2\). A(at least one woman), B(one man & one woman), C(no woman).
Ans. Total \(\binom{5}{2}=10\). \(n(C)=\binom{3}{2}=3\). \(n(B)=3\times 2=6\). \(n(A)=10-3=7\).
Q1(6). One coin & one die: A(Head & odd), B(Head or Tail with even), C(number >7 and tail).
Ans. \(n(S)=12\). \(n(A)=3\). \(n(B)=6\) (all even with either H or T). \(n(C)=0\) (no die outcome >7).
Practice Set 5.4 — Probability
Q1. Two coins: (1) \(P(\ge 1\ \text{head})\) (2) \(P(\text{no head})\).
Ans. \(\dfrac{3}{4}\) and \(\dfrac{1}{4}\).
Q2. Two dice: (1) \(P(\text{sum}\ge 10)\) (2) \(P(\text{sum}=33)\) (3) \(P(\text{first}>\text{second})\).
Ans. \(\dfrac{1}{6},\ 0,\ \dfrac{15}{36}=\dfrac{5}{12}\).
Q3. Tickets \(1\)–\(15\): \(P(\text{even})\), \(P(\text{multiple of }5)\).
Ans. \(\dfrac{7}{15}\) and \(\dfrac{3}{15}=\dfrac{1}{5}\).
Q4. Two-digit number from \(2,3,5,7,9\) (no repetition). \(P(\text{odd}), P(\text{multiple of }5)\).
Ans. Total \(20\). All are odd ⇒ \(1\). Multiples of \(5\): end with \(5\), tens \(\in\{2,3,7,9\}\Rightarrow \dfrac{4}{20}=\dfrac{1}{5}\).
Q5. From 52 cards: \(P(\text{ace})\), \(P(\text{spade})\).
Ans. \(\dfrac{1}{13}\) and \(\dfrac{1}{4}\).
Problem Set 5 — Mixed Practice
Q1. Which number cannot represent a probability? (A) \( \tfrac{2}{3}\) (B) \(1.5\) (C) \(15\%\) (D) \(0.7\)
Ans. \(1.5\) (greater than \(1\)).
Q2. Die rolled. \(P(\text{number}<3)\)?
Ans. \(\dfrac{2}{6}=\dfrac{1}{3}\).
Q3. \(P(\text{a randomly chosen number from }1\text{ to }100\ \text{is prime})\).
Ans. \(\dfrac{25}{100}=\dfrac{1}{4}\) (there are 25 primes up to 100).
Q4. Card \(1\)–\(40\). \(P(\text{multiple of }5)\).
Ans. \(5,10,15,20,25,30,35,40\Rightarrow \dfrac{8}{40}=\dfrac{1}{5}\).
Q5. If \(n(A)=2\) and \(P(A)=\tfrac{1}{5}\), find \(n(S)\).
Ans. \(n(S)=10\).
Q6. Basketball success probabilities: John \(=\tfrac{4}{5}=0.8\), Vasim \(=0.83\), Akash \(=0.58\). Who is greatest?
Ans. Vasim (\(0.83\)).
Q7. Hockey team: 6 defenders, 4 offenders, 1 goalie. Random captain: \(P(\text{goalie})\), \(P(\text{defender})\).
Ans. Total \(11\). \(P(\text{goalie})=\dfrac{1}{11}\). \(P(\text{defender})=\dfrac{6}{11}\).
Q8. 26 alphabet cards: \(P(\text{vowel})\).
Ans. Vowels \(=5\Rightarrow \dfrac{5}{26}\).
Q9. Balloons: 2 red, 3 blue, 4 green. Find probabilities.
Ans. Total \(9\). \(P(R)=\dfrac{2}{9}\), \(P(B)=\dfrac{3}{9}=\dfrac{1}{3}\), \(P(G)=\dfrac{4}{9}\).
Q10. Pens: \(5\) red, \(8\) blue, \(3\) green. \(P(\text{blue})\).
Ans. Total \(16\). \(P=\dfrac{8}{16}=\dfrac{1}{2}\).
Q11. Special die with faces A,B,C,D,E,A. \(P(A)\), \(P(D)\).
Ans. \(P(A)=\dfrac{2}{6}=\dfrac{1}{3}\); \(P(D)=\dfrac{1}{6}\).
Q12. Tickets \(1\)–\(30\). \(P(\text{odd})\), \(P(\text{perfect square})\).
Ans. Odd \(=15/30=1/2\). Squares \(=\{1,4,9,16,25\}\Rightarrow 5/30=1/6\).
Q13. Garden \(77\times 50\) m with circular lake diameter \(14\) m. \(P(\text{towel falls in lake})\).
Ans. \(=\dfrac{\pi\cdot 7^2}{77\cdot 50}=\dfrac{49\pi}{3850}\approx 0.040\).
Q14. Spinner \(1\)–\(8\). Find \(P(8), P(\text{odd}), P(>2), P(<9)\).
Ans. \(\dfrac{1}{8}, \dfrac{1}{2}, \dfrac{3}{4}, 1\) respectively.
Q15. Cards \(0\)–\(5\). \(P(\text{natural})\), \(P(<1)\), \(P(\text{whole})\), \(P(>5)\).
Ans. Natural \(\{1..5\}\Rightarrow 5/6\). \(<1\Rightarrow \{0\}\Rightarrow 1/6\). Whole \(\{0..5\}\Rightarrow 1\). \(>5\Rightarrow 0\).
Q16. Balls: 3 red, 3 white, 3 green. \(P(R)\), \(P(\text{not }R)\), \(P(R\text{ or }W)\).
Ans. \(\dfrac{1}{3}, \dfrac{2}{3}, \dfrac{2}{3}\).
Q17. “MATHEMATICS” cards (11). \(P(\text{card is }m)\).
Ans. \(\dfrac{2}{11}\).
Q18. 200 students; 135 like Kabaddi. \(P(\text{doesn’t like})\).
Ans. \(\dfrac{65}{200}=\dfrac{13}{40}\).
Q19. Two-digit number from \(0,1,2,3,4\) (repetition allowed). \(P(\text{prime})\), \(P(\text{multiple of }4)\), \(P(\text{multiple of }11)\).
Ans. Total \(20\). Prime \(=\dfrac{6}{20}=\dfrac{3}{10}\). Multiple of \(4\) \(=\dfrac{6}{20}=\dfrac{3}{10}\). Multiple of \(11\) \(=\dfrac{4}{20}=\dfrac{1}{5}\).
Q20. Die faces \(0\)–\(5\) rolled twice. \(P(\text{product }=0)\).
Ans. \(1\) zero appears with probability \(1-\left(\dfrac{5}{6}\right)^2=\dfrac{11}{36}\).
Q21. (Activity) Class spectacles: If \(n(S)\) total students and \(n(A)\) wear spectacles, write \(P(A)\) and \(P(\overline{A})\).
Ans. \(P(A)=\dfrac{n(A)}{n(S)}\), \(P(\overline{A})=1-\dfrac{n(A)}{n(S)}\).
🔑 Quick Summary
- Sample space: list all equally likely outcomes; count with \(n(S)\).
- Event: favourable outcomes; \(P(E)=\dfrac{n(E)}{n(S)}\).
- Ranges: \(0\le P(E)\le 1\); complement \(P(\overline{E})=1-P(E)\).
- Cards: \(52\) total; red \(26\); per suit \(13\); face \(12\); aces \(4\).
- Two dice: ordered pairs, \(36\) outcomes; use counting for sums/relations.