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Example 31 - If a fair coin is tossed 10 times, find probability

Example 31 - Chapter 13 Class 12 Probability - Part 2
Example 31 - Chapter 13 Class 12 Probability - Part 3 Example 31 - Chapter 13 Class 12 Probability - Part 4 Example 31 - Chapter 13 Class 12 Probability - Part 5


Transcript

Example 31 If a fair coin is tossed 10 times, find the probability of (i) exactly six heads (ii) at least six heads (iii) at most six headsIf a trial is Bernoulli, then There is finite number of trials They are independent Trial has 2 outcomes i.e. Probability success = P then Probability failure = q = 1 – P (4) Probability of success (P) is same for all trials Let X : Number of heads appearing Coin toss is a Bernoulli trial So, X has a binomial distribution P(X = x) = nCx 𝒒^(π’βˆ’π’™) 𝒑^𝒙 n = number of coins tosses = 10 p = Probability of head = 1/2 q = 1 – p = 1 – 1/2 = 1/2 Hence, P(X = x) = 10Cx (1/2)^π‘₯ (1/2)^(10βˆ’π‘₯) P(X = x) = 10Cx (1/2)^(10 βˆ’ π‘₯ + π‘₯) P(X = x) = 10Cx (𝟏/𝟐)^𝟏𝟎 Probability exactly six heads Probability exactly six heads = P(X = 6) Putting x = 6 in (1) P(X = 6) = 10C6 (1/2)^10 = (10 !)/((10 βˆ’ 6) ! Γ—6 !) Γ— (1/2)^10= (10 !)/(4 ! Γ— 6 !) Γ— 1/2^10 = 105/512 (ii) Probability appearing at least six heads i.e. P(X β‰₯ 6) P(X β‰₯ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 10C6 (1/2)^10 + 10C7 (1/2)^10 + 10C8 (1/2)^10 + 10C9 (1/2)^10 + 10C10 (1/2)^10 = (1/2)^10(10C6 + 10C7 + 10C8 + 10C9 + 10C10) = (1/2)^10(210 + 120 + 45 + 10 + 1) = (1/2)^10(386) = πŸπŸ—πŸ‘/πŸ“πŸπŸ (iii) Probability appearing at most six heads i.e. P(X ≀ 6) P(X ≀ 6) = P(X = 6) + P(X = 5) + P(X = 4) + P(X = 3) + P(X = 2) + P(X = 1) + P(X = 0) = 10C6 (1/2)^10 + 10C5 (1/2)^10 + 10C4 (1/2)^10 + 10C3 (1/2)^10 + 10C2 (1/2)^10 + 10C1 (1/2)^10+ 10C0 (1/2)^10 = (1/2)^10(10C6 + 10C5 + 10C4 + 10C3 + 10C2 + 10C1 + 10C0) = (1/2)^10(210 + 252 + 210 + 120 + 45 + 10 + 1) = (1/2)^10(848) = πŸ“πŸ‘/πŸ”πŸ’

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.