CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Probability

Q.5. An urn contains 5 white and 3 red balls. Find the probability distribution of the number of red balls, with replacements, in three draws.

Solution :

Let the event of drawing a red ball be R and X denote the discrete random variable “number of red balls” in a draw of 3 balls. Then X = 0, 1, 2, 3.
Here we have, P(R) = 3/8 and P(Not R) = 5/8.
P(X = 0) = P(Not R₁ not R₂ not R₃) = P(Not R1)P(Not R2)P(Not R3)
= 5/8×5/8×5/8 = 125/512.
P(X = 1) = P(Not R1 notR₂ R₃) + P(NotR₁ R₂ not R₃) + P(R₁ not R₂ not R₃) = 5/8×5/8×3/8 + 5/8×3/8×5/8 + 3/8×5/8×5/8 = 3(75/512) = 225/512.
P(X = 2) = P(Not R₁ R₂ R₃) + P(R₁ not R₂ R₃) + P(R₁ R₂ not R₃) = 5/8×3/8×3/8 + 3/8×5/8×3/8+ 3/8×3/8×5/8 = 3(45/521) = 135/512
P(X = 3) P(R1R2R3) = 3/8×3/8×3/8 = 27/512.
Hence the required probability distribution is

Q.6. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of number of success.

Solution :

We have, n = 4, total number of events = 6×6 = 36,
favourable evnts = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
p = P(doublet) = 6/36 = 1/6, q = 1 – p = 1 – 1/6 = 5/6.
Let x be the number of success
Therefore, x = 0, 1, 2, 3, 4.
As, P(x) = ⁿC_r q^{n – r} p^r
P(x = 0) = ⁴C₀(5/6)⁴(1/6)⁰ = (5/6)4 = 625/1296
P(x = 1) = ⁴C₁(5/6)³(1/6)¹ = 4(5/6)3(1/6) = 500/1296
P(x = 2) = ⁴C₂(5/6)2(1/6)² = {(4.3)/(2.1)}(25/36)(1/36) = 150/1296
P(x = 3) = ⁴C₃(5/6)1(1/6)³ = 4(5/6)(1/216) = 20/1296
P(x = 4) = ⁴C₄(5/6)0(1/6)⁴ = (1/6)4 = 1/1296
Therefore, required probability distribution is

13.5. Binomial Distribution.

Q.1. The mean and variance of a binomial distributions are 4 and 4/3 respectively. Find the distribution and P(X ≥ 1).

Solution :

We have, mean = np = 4 ---------- (1)
and variance = npq = 4/3 ----------- (2)
Therefore, dividing (2) by (1) we get
npq/np = (4/3)/4 => q = 1/3.
Or, p = 1 – q = 1 – 1/3 = 2/3.
From (1), n(2/3) = 4 => n = 4×(3/2) = 6.
Therefore, Binomial distribution is (q + p)n = (1/3 + 2/3)6.
P(X ≥ 1) = 1 – P(X = 0)
= 1 – ⁶C₀ (1/3)⁶(2/3)⁰
= 1 – 1/729
= 728/729.[Ans.]

Paper By Mr. M.P.Keshari
Email Id : [email protected]
Ph No. : 09434150289