# Misc 5 - Chapter 13 Class 12 Probability

Last updated at Feb. 15, 2020 by Teachoo

Last updated at Feb. 15, 2020 by Teachoo

Transcript

Misc 5 An urn contains 25 balls of which 10 balls bear a mark 'X' and the remaining 15 bear a mark 'Y'. A ball is drawn at random from the urn, its mark is noted down and it is replaced. If 6 balls are drawn in this way, find the probability that (i) all will bear 'X' mark. (ii) not more than 2 will bear 'Y' mark. (iii) at least one ball will bear 'Y' mark. (iv) the number of balls with 'X' mark and 'Y' mark will be equal.Let X : Number of balls with mark ‘X’ Drawing a ball is a Bernoulli trial So, X has a binomial distribution P(X = x) = nCx 𝒒^(𝒏−𝒙) 𝒑^𝒙 Here, n = number of balls drawn = 6 p = Probability of getting ball with ‘X’ mark = 10/25 = 2/5 q = 1 – p = 1 – 2/5 = 3/5 Hence, P(X = x) = 6Cx (𝟐/𝟓)^𝒙 (𝟑/𝟓)^(𝟔 − 𝒙) Probability that all will bear 'X' mark. Probability all balls has ‘X’ mark = P(X = 6) Putting x = 6 in (1) P(X = 6) = 6C6 (2/5)^6 (3/5)^(6 −6) = 6C6 (2/5)^6 (3/5)^0 = 1 × (2/5)^6× 1 = (𝟐/𝟓)^𝟔 Probability that not more than 2 will bear 'Y' mark. P(not more than 2 bear Y) = P(6X, 0Y) + P(5X, 1Y) + P(4X, 2Y) = P(X = 6) + P(X = 5) + P(X = 4) = 6C6(2/5)^6 (3/5)^(6−6) "+ 6C5" (2/5)^5 (3/5)^(6−5)+"6C4" 〖 (2/5)〗^4 (3/5)^(6−4) = 6C6(2/5)^6 (3/5)^0 "+ 6C5" (2/5)^5 (3/5)^1+"6C4" 〖 (2/5)〗^4 (3/5)^2 = 1 × (2/5)^6 "×" (3/5)^0 "+ 6 ×" (2/5)^5 (3/5)+"15" 〖 (2/5)〗^4 (3/5)^2 = (2/4)^4 [(2/5)^2+"6 ×" (2/5)(3/5)+15(3/5)^2 ] = (2/4)^4 [4/25+36/25+135/25]=(2/5)^4 [175/25] =𝟕(𝟐/𝟓)^𝟒 (iii) Probability that at least one ball will bear 'Y' mark. P(atleast one bears ‘Y’) = 1 – P(no balls bear ‘Y’) = 1 – P(all ball bears ‘X’) = 1 – P(X = 6) = 1 – 6C6(2/5)^6 (3/5)^(6−6) = 1 – 6C6(2/5)^6 (3/5)^0 = 1 – 1 × (2/5)^6 × 1 = 1 – (𝟐/𝟓)^𝟔 (iv) Probability that number of balls with 'X' mark & 'Y' mark will be equal. So, we will have 3X & 3Y balls P(X & Y marked balls are equal) = P(X = 3) = 6C3 (2/5)^3 (3/5)^(6−3) = 6C3 (2/5)^3 (3/5)^3 = (6 × 5 × 4 × 3!)/(3! × 3 × 2 × 1)×8/125×27/125 = 5 × 4 × 8/125 × 27/125 = 𝟖𝟔𝟒/𝟑𝟏𝟐𝟓

Miscellaneous

Misc 1

Misc 2 Important

Misc 3

Misc 4 Deleted for CBSE Board 2021 Exams only

Misc 5 Important Deleted for CBSE Board 2021 Exams only You are here

Misc 6 Important Deleted for CBSE Board 2021 Exams only

Misc 7 Important

Misc 8 Important

Misc 9 Important Deleted for CBSE Board 2021 Exams only

Misc 10 Important

Misc 11 Important

Misc 12

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18

Misc 19

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.