How many coins?

Probability Level 4

A Kaboobly Dooist discovers an infinite pool of coins with $ 1 \$1 , $ 2 \$2 and $ 5 \$5 coins distributed uniformly in it.

He takes a coin at random from the pool and puts it in his bag.

He does so until the sum of the values of all coins in his bag is equal to or more than $ 200 \$200 .

If the expected number of $ 1 \$1 coins in the bag is n n , what is the value of ⌊ n ⌋ \left\lfloor n\right\rfloor ?


The answer is 25.

Agnishom Chattopadhyay by Agnishom Chattopadhyay , 22, India

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1 solution

Nicola Mignoni
Apr 23, 2019

I'm not completely sure about this. Given that P ( $ 1 ) = P ( $ 2 ) = P ( $ 5 ) = 1 3 \displaystyle \mathbb{P}(\$1)=\mathbb{P}(\$2)=\mathbb{P}(\$5)=\frac{1}{3} , than the average amount summed up at each iteration is μ = 1 3 ( 1 + 2 + 5 ) = 8 3 \displaystyle \mu=\frac{1}{3}(1+2+5)=\frac{8}{3} . So, the expected numer of iterations N N to get to 200 200 is E ( N ) = 200 μ = 75 \displaystyle \mathbb{E}(N)=\frac{200}{\mu}=75 . Since the distribution is uniform, E ( $ 1 ) = E ( $ 2 ) = E ( $ 5 ) = E ( N ) 3 = 25 \displaystyle \mathbb{E}(\$1)=\mathbb{E}(\$2)=\mathbb{E}(\$5)=\frac{\mathbb{E}(N)}{3}=25 . I suspect that E ( $ 1 ) \mathbb{E}(\$1) is slightly bigger though.

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