Probability Class 11 Notes

• An outcome is a single result of a random experiment.
• A trial is one performance of the experiment.
• Events are collections of one or more outcomes.

• An event is a subset of the sample space.
• A sure event is the entire sample space S; probability = 1.
• An impossible event is the empty set ∅; probability = 0.

• Simple (elementary) event: contains exactly one outcome.
• Compound event: contains two or more outcomes.
• Complementary event A': the event that A does not occur.
• Mutually exclusive events: cannot occur together;
• $A\cap B=\varnothing$
• .
• Exhaustive events: their union covers the entire sample space.

• Axiom 1:
• $0\le P(A)\le 1$
• for every event A.
• Axiom 2:
• $P(S)=1$
• (probability of the sure event is 1).
• Axiom 3: For mutually exclusive events,
• $P(A\cup B)=P(A)+P(B)$
• .
• Classical definition: if all n outcomes are equally likely and m are favourable,
• $P(A)=\frac{m}{n}$
• .

• Complementary event:
• $P(A^{\prime })=1-P(A)$
• .
• If A and B are mutually exclusive,
• $P(A\cap B)=0$
• and the formula simplifies to
• $P(A\cup B)=P(A)+P(B)$
• .
• For three events:
• $P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)$
• .

• Mutually exclusive events are NOT the same as independent events.
• For mutually exclusive events,
• $P(A\cup B)=P(A)+P(B)$
• — no intersection to subtract.
• If events are both mutually exclusive and exhaustive, their probabilities sum to 1.

• Odds in favour = m:(n − m) if P(A) = m/n.
• Odds against = (n − m):m.
• If odds in favour are a:b, then
• $P(A)=\frac{a}{a+b}$
• .
• Converting between probability and odds is a common JEE short-cut.

Example: Two dice are thrown. Find the probability that the sum is 7.

Solution: Total outcomes = 6 × 6 = 36. Favourable pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 outcomes. Probability = 6/36 = 1/6.

Example: From a pack of 52 cards, one card is drawn. Find the probability that it is either a king or a heart.

Solution: P(king) = 4/52, P(heart) = 13/52, P(king of hearts) = 1/52. By the addition theorem, P(king or heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.

Exercises (NCERT Q1–Q28)

• 1) A die is rolled. Let E be the event "die shows 4" and F be the event "die shows even number". Are E and F mutually exclusive?
• 2) A die is thrown. Describe the following events: (i) A: a number less than 7; (ii) B: a number greater than 7; (iii) C: a multiple of 3; (iv) D: a number less than 4; (v) E: an even number greater than 4; (vi) F: a number not less than 3. Also find A ∪ B, A ∩ B, B ∪ C, E ∩ F, D ∩ E, A − C, D − E, E ∩ F′, F′.
• 3) An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the events: A: the sum is greater than 8; B: 2 occurs on either die; C: the sum is at least 7 and a multiple of 3. Which pairs of these events are mutually exclusive?
• 4) Three coins are tossed once. Let A denote the event "three heads show", B denote "two heads and one tail show", C denote "three tails show" and D denote "a head shows on the first coin". Which events are (i) mutually exclusive? (ii) simple? (iii) compound?
• 5) Three coins are tossed. Describe (i) two events which are mutually exclusive; (ii) three events which are mutually exclusive and exhaustive; (iii) two events which are not mutually exclusive; (iv) two events which are mutually exclusive but not exhaustive; (v) three events which are mutually exclusive but not exhaustive.
• 6) Two dice are thrown. The events A, B and C are: A: getting an even number on the first die; B: getting an odd number on the first die; C: getting the sum of the numbers on the dice ≤ 5. Describe the events (i) A′, (ii) not B, (iii) A or B, (iv) A and B, (v) A but not C, (vi) B or C, (vii) B and C, (viii) A ∩ B′ ∩ C′.
• 7) Refer to question 6 above. State true or false (give reason for your answer): (i) A and B are mutually exclusive; (ii) A and B are mutually exclusive and exhaustive; (iii) A = B′; (iv) A and C are mutually exclusive; (v) A and B′ are mutually exclusive; (vi) A′, B′, C are mutually exclusive and exhaustive.
• 8) Which of the following can not be a valid assignment of probabilities for outcomes of sample space S = {ω₁, ω₂, ω₃, ω₄, ω₅, ω₆, ω₇}? (a) 0.1, 0.01, 0.05, 0.03, 0.01, 0.2, 0.6; (b) 1/7, 1/7, 1/7, 1/7, 1/7, 1/7, 1/7; (c) 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7; (d) −0.1, 0.2, 0.3, 0.4, −0.2, 0.1, 0.3; (e) 1/14, 2/14, 3/14, 4/14, 5/14, 6/14, 15/14.
• 9) A coin is tossed twice. What is the probability that at least one tail occurs?
• 10) A die is thrown. Find the probability of: (i) a prime number appears; (ii) a number ≥ 3 appears; (iii) a number ≤ 1 appears; (iv) a number > 6 appears; (v) a number < 6 appears.
• 11) A card is selected from a pack of 52 cards. (a) How many points are there in the sample space? (b) Calculate the probability that the card is an ace of spades. (c) Calculate the probability that the card is (i) an ace, (ii) a black card.
• 12) A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. Find the probability that the sum of the numbers that turn up is (i) 3, (ii) 12.
• 13) There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?
• 14) A fair coin is tossed four times, and a person wins Re 1 for each head and loses Rs 1.50 for each tail that turns up. From the sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.
• 15) Three coins are tossed once. Find the probability of getting (i) 3 heads, (ii) 2 heads, (iii) at least 2 heads, (iv) at most 2 heads, (v) no head, (vi) 3 tails, (vii) exactly two tails, (viii) no tail, (ix) at most two tails.
• 16) If 2/11 is the probability of an event A, what is the probability of the event 'not A'?
• 17) A letter is chosen at random from the word 'ASSASSINATION'. Find the probability that letter is (i) a vowel, (ii) a consonant.
• 18) In a lottery, a person chooses six different natural numbers at random from 1 to 20, and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize? [Hint: order of the numbers is not important.]
• 19) Check whether the following probabilities P(A) and P(B) are consistently defined: (i) P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6; (ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8.
• 20) Fill in the blanks in the table: (i) P(A) = 1/3, P(B) = 1/5, P(A ∩ B) = 1/15, P(A ∪ B) = ?; (ii) P(A) = 0.35, P(B) = ?, P(A ∩ B) = 0.25, P(A ∪ B) = 0.6; (iii) P(A) = 0.5, P(B) = 0.35, P(A ∩ B) = ?, P(A ∪ B) = 0.7.
• 21) Given P(A) = 3/5 and P(B) = 1/5. Find P(A or B), if A and B are mutually exclusive events.
• 22) If E and F are events such that P(E) = 1/4, P(F) = 1/2 and P(E and F) = 1/8, find (i) P(E or F), (ii) P(not E and not F).
• 23) Events E and F are such that P(not E or not F) = 0.25. State whether E and F are mutually exclusive.
• 24) A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine (i) P(not A), (ii) P(not B), (iii) P(A or B).
• 25) In Class XI of a school, 40% of the students study Mathematics and 30% study Biology. 10% of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology.
• 26) In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both?
• 27) The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of passing the English examination is 0.75, what is the probability of passing the Hindi examination?
• 28) In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that (i) the student opted for NCC or NSS; (ii) the student has opted neither NCC nor NSS; (iii) the student has opted NSS but not NCC.

Answers at a glance

1. 1
   Q1 → No — not mutually exclusive, since 4 belongs to both.
2. 2
   Q2 → A = S, B = ∅, C = {3,6}, D = {1,2,3}, E = {6}, F = {3,4,5,6}; A∪B = S, A∩B = ∅, B∪C = {3,6}, E∩F = {6}, D∩E = ∅, A−C = {1,2,4,5}, D−E = {1,2,3}, E∩F′ = {6}, F′ = {1,2,5,6}.
3. 3
   Q3 → Only the pair (B, C) is mutually exclusive.
4. 4
   Q4 → (i) (A,B), (A,C), (B,C), (C,D); (ii) A and C; (iii) B and D.
5. 5
   Q5 → Any valid example is acceptable; the ones above use heads/tails patterns as listed.
6. 6
   Q6 → A′ = B; not B = A; A or B = S; A and B = ∅; A but not C = 14 listed outcomes; B and C = 6 listed outcomes; A∩B′∩C′ equals (v).
7. 7
   Q7 → (i) T, (ii) T, (iii) T, (iv) F, (v) F, (vi) F — reasons above.
8. 8
   Q8 → Invalid: (c), (d), (e). Valid: (a), (b).
9. 9
   Q9 → 3/4.
10. 10
    Q10 → (i) 1/2, (ii) 2/3, (iii) 1/6, (iv) 0, (v) 5/6.
11. 11
    Q11 → (a) 52, (b) 1/52, (c)(i) 1/13, (c)(ii) 1/2.
12. 12
    Q12 → (i) 1/12, (ii) 1/12.
13. 13
    Q13 → 3/5.
14. 14
    Q14 → Five amounts: Rs −6 (P 1/16), −3.5 (1/4), −1 (3/8), 1.5 (1/4), 4 (1/16).
15. 15
    Q15 → (i) 1/8, (ii) 3/8, (iii) 1/2, (iv) 7/8, (v) 1/8, (vi) 1/8, (vii) 3/8, (viii) 1/8, (ix) 7/8.
16. 16
    Q16 → 9/11.
17. 17
    Q17 → (i) 6/13, (ii) 7/13.
18. 18
    Q18 → 1/38760.
19. 19
    Q19 → (i) Not consistent; (ii) consistent.
20. 20
    Q20 → (i) 7/15, (ii) 0.5, (iii) 0.15.
21. 21
    Q21 → 4/5.
22. 22
    Q22 → (i) 5/8, (ii) 3/8.
23. 23
    Q23 → Not mutually exclusive (P(E ∩ F) = 0.75).
24. 24
    Q24 → (i) 0.58, (ii) 0.52, (iii) 0.74.
25. 25
    Q25 → 0.6 (i.e. 60%).
26. 26
    Q26 → 0.55.
27. 27
    Q27 → 0.65.
28. 28
    Q28 → (i) 19/30, (ii) 11/30, (iii) 2/15.

MCQ Questions

1 Mark Questions

1. 1
   Write the sample space of tossing a coin twice. (Ans: {HH, HT, TH, TT})
2. 2
   State the probability of a sure event. (Ans: 1)
3. 3
   If P(A) = 0.3, find P(A′). (Ans: 0.7)
4. 4
   How many outcomes are there when a die and a coin are tossed together? (Ans: 12)

2 Mark Questions

1. 1
   A card is drawn from a well-shuffled pack of 52 cards. Find the probability of a king. (Ans: 4/52 = 1/13)
2. 2
   Find the probability of getting an odd number when a die is rolled. (Ans: 3/6 = 1/2)
3. 3
   Two coins are tossed; find the probability of at least one head. (Ans: 3/4)

3 Mark Questions

1. 1
   Two dice are thrown; find the probability of getting a sum of 7. (Ans: 6/36 = 1/6)
2. 2
   A bag contains 4 red and 6 blue balls; two balls are drawn without replacement. Find the probability that both are red. (Ans: (4/10)(3/9) = 2/15)
3. 3
   If P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2, find P(A ∪ B). (Ans: 0.7)

4 Mark Questions

1. 1
   Prove the addition rule for two events P(A ∪ B) = P(A) + P(B) − P(A ∩ B) by counting elements, and apply it when P(A) = 0.6, P(B) = 0.5 and P(A ∪ B) = 0.9 to find P(A ∩ B). (Ans: 0.2)
2. 2
   Three coins are tossed. Write the full sample space, then find the probability of (i) exactly two heads, (ii) at least two heads, (iii) no heads, examining the distribution of the number of heads. (Ans: (i) 3/8, (ii) 4/8 = 1/2, (iii) 1/8)