Using Conditional probability,
The probability that after a fair coin is tossed 5 times that the result will contain 3 'heads' given that the first toss was a 'head' is 3/5.
A fair coin is tossed five times .
so, total possible outcomes = 2⁵ = 32
Let A: head observed on first toss
B: 3 heads observed.
Probability that the first toss was head (p) = 1/2
we have to find out probability that on 5 tosses result contain 3 heads given that first toss wss head .
Using Binomial Probability distribution
P (X= x) = ⁿCₓ (p)ˣ (q)⁽ⁿ⁻ˣ⁾
x = 3 , n = 5
we get , P(B) = P(X= 3 ) = ⁵C₃(1/2)³(1/2)²
=> P(B) = 10 ×(1/2)⁵= 5/16
since the first toss is fixed as a head.
P(A ∩ B) =P( X₁ = 1 and X₂+ X₃ + X₄ + X₅) = P(X₁= 1)P(X₂+X₃+X₄+X₅)
= 1/2 × ⁴C₂ (1/2)⁵ = 3 (1/2)⁴
The required probability is , Conditional probability P(B/A) = P(A ∩ B)/P(A)
= 3(1/2)⁴/ 5/16 = 3/5
Hence , the required probability is 3/5.
To learn more about Conditional Probability, refer:
https://brainly.com/question/10567654
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