| Class XI |
Mathematics |
Mathematics |
13 |
EXERCISE 13.2 Find the mean and variance for each of the data in Exercies 1 to 5. 1. 6,7,10,12,13,4,8,12 2. First n natural numbers 3. First10multi... |
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| Class XI |
Mathematics |
Mathematics |
13 |
2 1 2 Variance ( ) = ∑ (xi− x) , i.e.,= 20 ∑ (i − x) n i=1 i=1 (x − x)2 or i=1 i =100 If each observation is multiplied by 2, and the new resulting... |
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| Class XI |
Mathematics |
Mathematics |
13 |
+ 38.72 Therefore x + y = 97 ... (2) But from (1), we have x + y + 2xy = 169 ... (3) From (2) and (3), we have 2xy = 72 ... (4) Subtracting (4) fro... |
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| Class XI |
Mathematics |
Mathematics |
13 |
1 i=1 Thus,thevarianceofthenewobservationsissameasthatoftheoriginalobservations. Note We may note that adding (or subtracting) a positive number to... |
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| Class XI |
Mathematics |
Mathematics |
13 |
deviation of a continuous frequency distribution 2 1 2 1 2 2 σ = N ∑ fi( i x ) , σ = N N∑ i i −(∑ fi i fi Shortcut method to find variance and stand... |
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| Class XI |
Mathematics |
Mathematics |
14 |
Chapter 14 PROBABILITY v Where a mathematical reasoning can be had, it is as great a folly to make use of any other, as to grope for a thing in the... |
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| Class XI |
Mathematics |
Mathematics |
14 |
less than 4 appears”. If actually ‘1’ had appeared on the die then we say that event E has occurred.As a matter of fact if outcomes are 2 or 3, we ... |
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| Class XI |
Mathematics |
Mathematics |
14 |
1 {HH}, E ={2T}, E = { 3H} and E ={TT}.4 Reprint 2026-27 PROBABILITY 291 3. Compound Event If an event has more than one sample point, it is called... |
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| Class XI |
Mathematics |
Mathematics |
14 |
B are two events associated with a sample space, then ‘A∪ B’is the event ‘eitherAor B or both’.This event ‘A∪ B’is also called ‘Aor B’. Therefore E... |
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| Class XI |
Mathematics |
Mathematics |
14 |
= {1, 2, 3, 5} (ii) ‘A and B’=A ∩ B = {3,5} (iii) ‘A but not B’= A – B = {2} (iv) ‘not A’=A′ = {1,4,6} 14.1.4 Mutually exclusive events In the expe... |
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| Class XI |
Mathematics |
Mathematics |
14 |
3} ∪ {3, 4} ∪ {5, 6} = S. Such eventsA, B and C are called exhaustive events. In general, if E , E ,1...2 E arenn events of a sample space S and if... |
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| Class XI |
Mathematics |
Mathematics |
14 |
find that A ∩ B = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 6)} ≠ φ Therefore, Aand B are not mutually exclusive events. Similarly A ∩ C ≠ φ, A ... |
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| Class XI |
Mathematics |
Mathematics |
14 |
B, A∩ B, B ∪ C, E ∩ F, D ∩ E, A – C, D – E, E ∩ F′, F′ Reprint 2026-27 PROBABILITY 295 3. An experiment involves rolling a pair of dice and recordi... |
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| Class XI |
Mathematics |
Mathematics |
14 |
are mutually exclusive (v) A and B ′ are mutually exclusive. (vi) A ′, ′, C are mutually exclusive and exhaustive. 14.2 AxiomaticApproachtoProbabil... |
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| Class XI |
Mathematics |
Mathematics |
14 |
i Note It may be noted that the singleton {ω} is called elementary event and A i for notational convenience, we write P(ω )ifor P({ω }i. For exampl... |
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| Class XI |
Mathematics |
Mathematics |
14 |
(e) 0.1 0.2 0.3 0.4 0.5 0.6 Solution (a) Condition (i): Each of the numier p(ω) is positive and less than one. Condition(ii):Sumofprobabilities 1 1... |
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| Class XI |
Mathematics |
Mathematics |
3 |
59 2 2 Since P 1 3 = P 2 4 we have P P 1 3 P . 2 4 Therefore, 2 –2 (cos x cos y – sin x sin y) = 2 – 2 cos (x + y). Hence cos (x + y) = cos x cos y... |
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| Class XI |
Mathematics |
Mathematics |
14 |
S = {HH, HT, TH, TT} Letthefollowingprobabilitiesbeassignedtotheoutcomes 1 1 2 9 P(HH) = , P(HT) = , P(TH) = , P(TT) = 4 7 7 28 Clearly this assign... |
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| Class XI |
Mathematics |
Mathematics |
14 |
= 2 Also P(A) = P(HHT) + P(HTH) + P(THH) = 8 Reprint 2026-27 300 MATHEMATICS and P(B) = P(HTH) + P(THH) + P(HHH) = 3 Therefore P(A) + P(B) = 3 + 3=... |
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| Class XI |
Mathematics |
Mathematics |
14 |
(B – A), where A ∩ B and B – A are mutually exclusive. Using Axiom(iii)ofprobability,weget Reprint 2026-27 PROBABILITY 301 P (A ∪B) = P (A) + P (B ... |
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| Class XI |
Mathematics |
Mathematics |
14 |
P(A) A ′ We now consider some examples and exercises having equally likely outcomes unless stated otherwise. Example 5 One card is drawn from a wel... |
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| Class XI |
Mathematics |
Mathematics |
14 |
Now P(E ∪ F) = P(E) + P(F) – P(E ∩ F) or P(E ∪ F) = 0.05 + 0.10 – 0.02 = 0.13 Therefore P(E ´ ∩ F ´) = P(E ∪ F)´ = 1 – P(E ∪ F) = 1 – 0.13 = 0.87 (... |
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| Class XI |
Mathematics |
Mathematics |
14 |
man =) 1 1= = 4C 2 2×3 3 (c) Two men can be selected in 2C way. C 2 1 1 Hence P (wo men = ) 4C = 4C = 6 2 2 EXERCISE14.2 1. Whichofthefollowingcann... |
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| Class XI |
Mathematics |
Mathematics |
14 |
is tossed four times, and a person win Re 1 for each head and lose Rs 1.50 for each tail that turns up. Fromthesamplespacecalculatehowmanydifferent... |
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| Class XI |
Mathematics |
Mathematics |
14 |
= 0.48 and P(A and B) = 0.16. Determine (i) P(not A), (ii) P(not B) and (iii) P(A or B) 18. In Class XI of a school 40% of the students study Mathe... |
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