Green and blue gummy bears

Probability Level 3

A bag contains gummy bears of 2 colors: blue and green.
There are 6 blue gummy bears.
If 2 bears are drawn at random from the bag, the probability that there is a bear of each colour is 1 2 \dfrac12 .
How many green gummy bears are in the bag?
Give the sum of all possible answers.

Image Credit: Flickr Derrick Diemont .


The answer is 13.

Paul Fournier by Paul Fournier , 70, Canada

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

Suppose that there are n n green gummy bears. Now we require that the sum of the probabilities of picking a green gummy bear followed by a blue one and that of picking a blue gummy bear followed by a green one add to 1 / 2 , 1/2, i.e., that

( n n + 6 ) ( 6 n + 5 ) + ( 6 n + 6 ) ( n n + 5 ) = 1 2 \left(\dfrac{n}{n + 6}\right)\left(\dfrac{6}{n + 5}\right) + \left(\dfrac{6}{n + 6}\right)\left(\dfrac{n}{n + 5}\right) = \dfrac{1}{2}

⟹ 24 n = ( n + 5 ) ( n + 6 ) ⟹ n 2 − 13 n + 30 = 0 ⟹ ( n − 3 ) ( n − 10 ) = 0. \Longrightarrow 24n = (n + 5)(n + 6) \Longrightarrow n^{2} - 13n + 30 = 0 \Longrightarrow (n - 3)(n - 10) = 0.

So n n can equal either 3 3 or 10 , 10, the sum of which is 13 . \boxed{13}.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

i used ( 6 1 ) ∗ ( n 1 ) ( 6 + n 2 ) = 1 2 \frac{ {6 \choose 1} * {n \choose 1}}{ 6+n \choose 2 } = \frac{1}{2}

ADRABI Abderrahim - 5 years, 9 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...