Isn't 100000 100000 such a huge number (Part-7)?

Probability Level 4

100 100 identical coins each with probability P P of coming Heads are tossed simultaneously. If 0 < P < 1 0<P<1 and the probability of Heads coming on 50 50 coins is equal to that of Heads coming on 51 51 coins, then the value of P P can be expressed as a b \dfrac{a}{b} .

Find the value of 100000 × ( a + b ) 100000\times(a+b) .


The answer is 15200000.

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Yash Singhal by Yash Singhal , 22, India

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1 solution

Pranjal Jain
Feb 17, 2015

Probability of getting 50 50 heads= ( 100 50 ) P 50 ( 1 − P ) 50 \dbinom{100}{50}P^{50}(1-P)^{50}

Probability of getting 51 51 heads= ( 100 51 ) P 51 ( 1 − P ) 49 \dbinom{100}{51}P^{51}(1-P)^{49}

Equating both expressions,

x 1 − x = ( 100 50 ) ( 100 51 ) = 51 50 ⇒ x = 51 101 \dfrac{x}{1-x}=\dfrac{\dbinom{100}{50}}{\dbinom{100}{51}}=\dfrac{51}{50}\\\Rightarrow x=\dfrac{51}{101}

a = 51 , b = 101 ⇒ a + b = 152 a=51,b=101\Rightarrow a+b=152

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Perfect! Upvoted+1

Yash Singhal - 6 years ago

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