# Example 29

Last updated at Dec. 18, 2018 by Teachoo

Last updated at Dec. 18, 2018 by Teachoo

Transcript

Example 29 Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings. Let X be the number of kings obtained We can get 0, 1, or 2 kings So, value of X is 0, 1 or 2 Total number of ways to draw 2 cards out of 52 is Total ways = 52C2 = 1326 P(X = 0) i.e. probability of getting 0 kings Number of ways to get 0 kings = Number of ways to select 2 cards out of non king cards = Number of ways to select 2 cards out of (52 – 4) 48 cards = 48C2 = 1128 P(X = 0) = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 0 𝑘𝑖𝑛𝑔𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠) = 1128/1326 P(X = 1) i.e. probability of getting 1 kings Number of ways to get 1 kings = Number of ways to select 1 king out of 4 king cards × Number of ways to select 1 card from 48 non king cards = 4C1 × 48C1 = 4 × 48 = 192 P(X = 1) = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 1 𝑘𝑖𝑛𝑔)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠) = 192/1326 P(X = 2) i.e. probability of getting 2 kings Number of ways to get 1 kings = Number of ways of selecting 2 kings out of 4 king cards = 4C2 = 6 P(X = 2) = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑔𝑒𝑡 2 𝑘𝑖𝑛𝑔𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠) = 6/1326 The probability distribution is The expectation value E(x) is given by 𝜇="E(X)"=∑2_(𝑖 = 1)^𝑛▒𝑥𝑖𝑝𝑖 = 0 × 1128/1326 +"1 ×" 192/1326 + 2 × 6/1326 = 0 + (192 + 12 )/1326 = 204/1326 = 34/221 = 2/13 The variance of x is given by : Var (𝑋)=𝐸(𝑋^2 )−[𝐸(𝑋)]^2 So, finding 𝐸(𝑋^2 ) E(𝑋^2 )=∑2_(𝑖 = 1)^𝑛▒〖〖𝑥_𝑖〗^2 𝑝𝑖〗 = 02 × 1128/1326+"12 × " 192/1326+ 22 × 6/1326 = 0+(192 + 4 × 6)/1326 = (192 + 24)/1326 = 216/1326 = 36/221 Now, Var (𝑿)=𝑬(𝑿^𝟐 )−[𝑬(𝑿)]^𝟐 = 36/221−(34/221)^2 = 1/221 [36−〖34〗^2/221] = 1/221 [(221 × 36 − 1156)/221] = 6800/(221)^2 ∴ Variance var (𝑋) = 6800/(221)^2 Standard deviation is given by 𝝈_𝒙=√(𝑣𝑎𝑟(𝑋) ) = √(6800/(221)^2 ) = √6800/221 = 82.46/221 = 8246/22100 = 0.37

Chapter 13 Class 12 Probability

Example 7
Important

Example 6 Important

Ex 13.1, 10 Important

Ex 13.1, 12 Important

Example 11 Important

Ex 13.2, 7 Important

Ex 13.2, 11 Important

Ex 13.2, 14 Important

Example 17 Important

Example 18 Important

Example 20 Important

Example 21 Important

Ex 13.3, 2 Important

Ex 13.3, 4 Important

Ex 13.3, 8 Important

Ex 13.3, 10 Important

Ex 13.3, 12 Important

Ex 13.3, 13 Important

Example 25 Important

Example 26 Important

Example 27 Important

Example 28 Important

Example 29 Important You are here

Ex 13.4, 3 Important

Ex 13.4, 6 Important

Ex 13.4, 11 Important

Ex 13.4, 15 Important

Example 31 Important

Example 32 Important

Ex 13.5, 4 Important

Ex 13.5, 6 Important

Ex 13.5, 10 Important

Ex 13.5, 13 Important

Example 35 Important

Example 36 Important

Misc 6 Important

Misc 9 Important

Misc 11 Important

Misc 13 Important

Misc 16 Important

Class 12

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
- Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.