10 Boxes

Probability Level 2

There are 10 boxes containing blue and red balls.

The number of blue balls in the $n^\text{th}$ box is given by $B(n) = 2^n$ .
The number of red balls in the $n^\text{th}$ box is given by $R(n) = 1024 - B(n)$ .

A box is picked at random, and a ball is chosen randomly from that box. If the ball is blue, and the probability that the $10^\text{th}$ box was picked can be expressed as $\frac ab$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

The answer is 1535.

2 solutions

Geoff Pilling
May 10, 2016

Let A = probability that the 10th box was picked

Let B = initial probability that a blue ball was chosen.

Using Baye's theorem ,

$P(A|B) = \frac{P(B|A)*P(A)}{P(B)} = \frac{(1)*(1/10)}{(1/10)*\frac{2^1+2^2+...+2^{10}}{1024}} = \frac{1024}{2^1+2^2+...+2^{10}} = \frac{1024}{2046} = \frac{512}{1023}$

And $512 +1023 = \boxed{1535}$

Why do u multiply the denominator by 1/10?

Ashish Sacheti - 4 years, 11 months ago

Its 1/10th times each of the probabilities for each box that it contains a blue ball. i.e. Another way of writing it would have been $P_1/10 + P_2/10 + ... + P_{10}/10$

Geoff Pilling - 4 years, 11 months ago

@Ashish Sacheti P(B) is probability of choosing a blue ball from a box. To choose a blue ball you need to 'choose a box' and 'pick a blue ball'

Probability to choose box = 1/10 Total number of balls in any box = 2^n + 1024 - 2^n = 1024 Probability of selecting blue ball from 1st box = 2^1/1024 Probability of selecting blue ball from 2nd box = 2^2/1024 .... Probability of selecting blue ball from 10th box = 2^10/1024

P(B) = (1/10) * ((2+4+8+...2^10) / 1024)

Mayank Prasad - 4 years, 9 months ago

Each box must contain 1024 balls only since the number of red balls = 1024 - number of blue balls => red balls + blue balls = 1024

Therefore, P(B) = Total Number of Blue Balls in 10 boxes / Total Number of Balls in all 10 boxes = ( 2^1 + 2^2 + …. 2^10 ) / 10 1024 = (Sum of 10 terms of a G.P. with r = 2 and a = 2 ) / 10 1024 = 2046 / 10*1024

Sanket Kale - 1 year, 12 months ago

I think you can also simplify the probability of blue to (number of blue balls total / number of balls total) since you have an equal chance of getting any blue ball. Am I wrong?

e r - 2 years, 6 months ago

Philip Norton
Dec 18, 2017

Because each box has equal numbers of balls, we can simplify this problem. The probability of the ball being from the 10th box is

$P(10th)=\frac{B(10)}{B(1)+B(2)+B(3)+...+B(10)}=\frac{1024}{2+4+8+...+1024}=\frac{1024}{2046}=\frac{512}{1023}$

And $512+1023=1535$

easy question

yash jamine - 11 months, 2 weeks ago

10 Boxes

The answer is 1535.

2 solutions

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