Prime Coins

Probability Level 3

John has a hobby of collecting coins. Currently he has 50 coins. Out of that first 10 are of face value 1 , next 10 of face value 2 , next ten of face value 5, next 10 of face value 7 , and the last 10 of face value 11 . He blindfolds himself and randomly picks one of these 50 coins , notes down the face value, and then returns the coin back to its original place. John keeps on repeating this process i.e. drawing one coin out of the 50 coins, until he has a total of 5 observations (of face values). If the probability that the maximum of these 5 readings is 7 can be expressed as a b \frac{a}{b} , where a and b are relatively prime, then find a + b a+b .


The answer is 3906.

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Utkarsh Grover by Utkarsh Grover , 22, India

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1 solution

Zico Quintina
May 4, 2018

In order for the maximum reading to be 7, John needs to draw no 11's but at least one 7. The probability of this is P[no 11 ’s] - P[no 7 ’s or 11 ’s ] = ( 4 5 ) 5 − ( 3 5 ) 5 = 781 3125 \text{P[no }11\text{'s] - P[no }7\text{'s or }11\text{'s}] = \left( \dfrac{4}{5} \right)^5 - \left( \dfrac{3}{5} \right)^5 = \dfrac{781}{3125} and the answer we're seeking is a + b = 781 + 3125 = 3906 a+b=781+3125=\boxed{3906}

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