A problem by Tianbo Chen

Level pending

The sum ∑ i = 0 100 ( x ( 100 x ) ( . 83 ) x ( . 17 ) 100 − x ) \sum_{i=0}^{100} (x \binom{100}{x}(.83)^x (.17)^{100-x}) is a two digit number. Find the sum of the digits.

10 11 9 8

Tianbo Chen by Tianbo Chen , 24, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Pi Han Goh
Dec 22, 2013

Motivation : Because ( 100 x ) ( 0.83 ) x ( 0.17 ) 100 − x {100 \choose x} (0.83)^x (0.17)^{100-x} looks like the probability of a binomial distribution with n = 100 n=100 trials and p = 0.83 p=0.83 success rate. Then the sum over all possible trials of the number of trials times its probability is simply the mean.

Hence,

∑ x = 0 100 ( x ( 100 x ) ( 0.83 ) x ( 0.17 ) 100 − x ) = ∑ x = 0 100 ( x ⋅ P r ( X = x ) ) = E ( X ) = n p = 100 × 0.83 = 83 \begin{aligned} \displaystyle \sum_{x=0}^{100} \left ( x {100 \choose x} (0.83)^x (0.17)^{100-x} \right ) & = & \sum_{x=0}^{100} \left ( x \cdot \mathrm{Pr}(X=x) \right ) \\ & = & \mathrm{E}(X) \\ & = & np \\ & = & 100 \times 0.83 \\ & = & 83 \\ \end{aligned}

The sum of digits is simply 8 + 3 = 11 8+3 = \boxed{11}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...