Class 12 Maths Probability – NCERT Formulas & Examples

Class 12 Maths Chapter 13 – Probability

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Introduction

Probability is the study of chance and uncertainty. It helps us quantify the likelihood of events and is crucial for statistics, science, finance, and engineering. This chapter includes conditional probability, Bayes theorem, random variables, expectation, and variance.

Formulas Used

• P(A) = Number of favorable outcomes / Total outcomes
• Complementary Rule: P(A') = 1 - P(A)
• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Multiplication Rule: P(A ∩ B) = P(A) * P(B|A)
• Conditional Probability: P(A|B) = P(A ∩ B)/P(B)
• Bayes Theorem: P(Ai|B) = P(Ai)P(B|Ai) / Σ P(Aj)P(B|Aj)
• Expectation: E(X) = Σ x·P(X=x)
• Variance: Var(X) = E(X²) - (E(X))²

Examples

1. Find the probability of getting at least one head in three tosses of a coin.
2. A bag contains 5 red and 3 black balls. One ball is drawn. Find the probability it is red.

30 NCERT Questions with Solutions

Q1. Tossing a die twice, find the probability of getting sum 7.

Solution: Total outcomes = 6×6 = 36. Favorable outcomes = (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6. Probability = 6/36 = 1/6.

Q2. A bag contains 5 red and 3 black balls. One ball is drawn. Find the probability it is black.

Solution: Total balls = 8. Favorable outcomes (black) = 3. Probability = 3/8.

Q3. Two coins are tossed simultaneously. Find the probability of getting at least one head.

Solution: Total outcomes = 4. Favorable outcomes = (H,H),(H,T),(T,H) = 3. Probability = 3/4.

Q4. A card is drawn from a deck. Find the probability it is a face card.

Solution: Face cards = J, Q, K of each suit = 12 cards. Total cards = 52. Probability = 12/52 = 3/13.

Q5. A die is rolled. Probability of getting an even number?

Solution: Even numbers = 2,4,6 = 3 outcomes. Total outcomes = 6. Probability = 3/6 = 1/2.

Q6. Two dice are thrown. Probability that both show the same number.

Solution: Favorable outcomes = (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) = 6. Total outcomes = 36. Probability = 6/36 = 1/6.

Q7. A bag has 7 white and 3 black balls. Two balls drawn at random. Probability both are white.

Solution: Total ways = 10C2 = 45. Favorable ways = 7C2 = 21. Probability = 21/45 = 7/15.

Q8. Probability of drawing a red card or a king from a deck.

Solution: Red cards = 26, Kings = 4, Red kings = 2. Probability = (26+4-2)/52 = 28/52 = 7/13.

Q9. One card drawn. Probability of neither king nor queen.

Solution: Non-face cards = 52-8 = 44. Probability = 44/52 = 11/13.

Q10. A coin is tossed 3 times. Probability of exactly two heads.

Solution: Outcomes with 2 heads: HHT, HTH, THH = 3. Total outcomes = 8. Probability = 3/8.

Q11. One card drawn. Probability it is red or a spade.

Solution: Red cards = 26, Spades = 13, Red Spades = 0. Probability = (26+13)/52 = 39/52 = 3/4.

Q12. A die is rolled twice. Probability sum is divisible by 3.

Solution: Favorable pairs: (1,2),(2,1),(3,3),(1,5),(5,1),(2,4),(4,2),(3,6),(6,3),(4,5),(5,4),(6,6)=12. Total=36. Probability=12/36=1/3.

Q13. A bag has 3 red, 4 blue, 5 green balls. Probability of blue or green.

Solution: Favorable=4+5=9, Total=12. Probability=9/12=3/4.

Q14. Two dice thrown. Probability one shows 2 and other shows 5.

Solution: Outcomes=(2,5),(5,2)=2. Total=36. Probability=2/36=1/18.

Q15. Probability of getting at least one tail in 3 coin tosses.

Solution: Only all heads=1 outcome. Total=8. Probability=1-1/8=7/8.

Q16. Bag contains 6 red, 5 black balls. Two drawn. Probability one red one black.

Solution: Favorable=6*5=30, Total=11C2=55. Probability=30/55=6/11.

Q17. Toss coin and die. Probability head and 4.

Solution: Outcomes=2*6=12. Favorable=1*1=1. Probability=1/12.

Q18. Draw card. Probability heart or queen.

Solution: Hearts=13, Queens=4, Queen of hearts=1. Probability=(13+4-1)/52=16/52=4/13.

Q19. Two dice rolled. Probability sum less than 5.

Solution: Pairs: (1,1),(1,2),(2,1),(1,3),(3,1),(2,2). Total=6. Total outcomes=36. Probability=6/36=1/6.

Q20. Bag has 4 red,3 green,2 black. Draw 1. Probability not black.

Solution: Favorable=4+3=7, Total=9. Probability=7/9.

Q21. Toss 2 coins. Probability exactly one head.

Solution: Outcomes=HH,HT,TH,TT. Exactly one head=HT,TH=2. Probability=2/4=1/2.

Q22. A die is rolled. Probability of not 6.

Solution: Not 6 outcomes=5. Total=6. Probability=5/6.

Q23. Draw card. Probability number card (2–10).

Solution: Number cards=9*4=36. Total=52. Probability=36/52=9/13.

Q24. Two dice rolled. Probability sum=8.

Solution: Pairs: (2,6),(6,2),(3,5),(5,3),(4,4)=5. Total=36. Probability=5/36.

Q25. Probability of getting head in single coin toss.

Solution: Favorable=1, Total=2. Probability=1/2.

Q26. Bag with 3 red, 2 green balls. Two drawn. Probability both same color.

Solution: Red=3C2=3, Green=2C2=1, Total favorable=4, Total=5C2=10. Probability=4/10=2/5.

Q27. Toss coin 4 times. Probability all tails.

Solution: Total=16, Favorable=1. Probability=1/16.

Q28. Two dice. Probability sum odd.

Solution: Sum odd if one odd one even. Outcomes=18. Total=36. Probability=1/2.

Q29. Draw card. Probability club or face card.

Solution: Clubs=13, Face cards=12, Face clubs=3. Probability=(13+12-3)/52=22/52=11/26.

Q30. Toss coin twice. Probability at most one head.

Solution: Outcomes: HH, HT, TH, TT. At most one head: HT, TH, TT=3. Probability=3/4.

FAQs on Probability

1. Q: What is conditional probability?
   A: P(A|B) = P(A ∩ B)/P(B).
2. Q: What is Bayes theorem used for?
   A: To update probabilities based on new evidence.
3. Q: What is a random variable?
   A: A variable representing numerical outcomes of a random phenomenon.
4. Q: Difference between discrete and continuous probability?
   A: Discrete has countable outcomes; continuous has uncountably infinite outcomes.
5. Q: How is expectation calculated?
   A: E(X) = Σ x·P(X=x).
6. Q: How is variance calculated?
   A: Var(X) = E(X²) - (E(X))².

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