# Example 29 - Chapter 13 Class 12 Probability

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 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 𝑘𝑖𝑛𝑔𝑠𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 11281326 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 𝑘𝑖𝑛𝑔𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 1921326 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 𝑘𝑖𝑛𝑔𝑠𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 61326 The probability distribution is The expectation value E(x) is given by 𝜇=E(X)= 𝑖 = 1𝑛𝑥𝑖𝑝𝑖 = 0 × 11281326 +1 × 1921326 + 2 × 61326 = 0 + 192 + 12 1326 = 2041326 = 34221 = 213 The variance of x is given by : Var 𝑋=𝐸 𝑋2− 𝐸 𝑋2 So, finding 𝐸 𝑋2 E 𝑋2= 𝑖 = 1𝑛 𝑥𝑖2𝑝𝑖 = 02 × 11281326+12 × 1921326+ 22 × 61326 = 0+ 192 + 4 × 61326 = 192 + 241326 = 2161326 = 36221 Now, Var 𝑿=𝑬 𝑿𝟐− 𝑬 𝑿𝟐 = 36221− 342212 = 1221 36− 342221 = 1221 221 × 36 − 1156221 = 6800 2212 ∴ Variance var 𝑋 = 6800 2212 Standard deviation is given by 𝝈𝒙= 𝑣𝑎𝑟 𝑋 = 6800 2212 = 6800221 = 0.37

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Example 29 Important You are here

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Example 31 Important

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Example 37

Chapter 13 Class 12 Probability

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.