Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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A Biased Coin with Probability P, 0 < P < 1, of Heads is Tossed Until a Head Appears for the First Time. If the Probability that the Number of Tosses Required is Even is 2/5, Then P Equals (A) 1/3 - Mathematics

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Question

A biased coin with probability p, 0 < p < 1, of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is 2/5, then p equals

Options
  • 1/3

  • 2/3

  • 2/5

  • 3/5

     

Solution

1/3
Probability of heads = p
Probability of Tails = (1 − p)
Probability that first head appears at even turn
Pe = (1 − p) p + (1 − p)3p+(1 − p)5p + .....
     = (1 − p) p (1 + (1 − p)2 + (1 − p)4+ .....)

\[= \left( 1 - p \right)p\left( \frac{1}{1 - \left( 1 - p \right)^2} \right)\]
\[ = \left( 1 - p \right)p\left( \frac{1}{- p^2 + 2p} \right)\]

\[P_e = \frac{\left( 1 - p \right)}{\left( 2 - p \right)}\]
\[ P_e = \frac{2}{5}\]
\[\frac{1 - p}{2 - p} = \frac{2}{5}\]
\[5 - 5p = 4 - 2p\]
\[3p = 1\]
\[p = \frac{1}{3}\]

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APPEARS IN

 RD Sharma Solution for Mathematics for Class 12 (Set of 2 Volume) (2018 (Latest))
Chapter 33: Binomial Distribution
MCQ | Q: 13 | Page no. 28
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A Biased Coin with Probability P, 0 < P < 1, of Heads is Tossed Until a Head Appears for the First Time. If the Probability that the Number of Tosses Required is Even is 2/5, Then P Equals (A) 1/3 Concept: Bernoulli Trials and Binomial Distribution.
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