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### What If You Paint the Balls?

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Date: 03/25/2002 at 05:31:17
From: John Midgley
Subject: Probability - what if you *paint* the balls?

Hi,

You have 800 blue balls in a bag. You take out 35 of them, paint them
red, and replace them. Do this 23 times and you *might* paint all of
them - what are the odds [1]? As the number of repetitions increases,
how can you model the improving odds of painting all the balls?

Alternatively, is there an *intelligent* question I should be asking

I'm stuck because:
a) I should have been paying attention when they tried to teach me;
b) The damned things change colour.

Incidentally, if Dr Math had been around before those 240 months
elapsed, I *might* have passed 'A' level maths (I'm in England).
[1] About the same probability as in the sentence above.

Regards,
John Midgley
```

```
Date: 03/25/2002 at 08:31:50
From: Doctor Anthony
Subject: Re: Probability - what if you *paint* the balls?

We can use a difference equation to represent what is happening.
Every painting operation covers 35/800 of the total.

Let u(n) be the number that are red after n painting turns.

800-u(n) will be the number that are still blue.

Then  u(n+1) = u(n) + (35/800)[800-u(n)]

u(n+1) = u(n)[1 - 35/800] + 35

u(n+1) = (153/160).u(n) + 35

u(n+1) - (153/160).u(n) = 35

So we must solve the difference equation to find u(n) in terms of n.

Let u(n) = v(n) + w(n) where v(n) is the complementary function and
w(n) the particular solution.

The complementary function is   x - 153/160 = 0 giving

x = 153/160 so C.F. is    v(n) = A.(153/160)^n

For the particular solution we assume

w(n) = a.n + b  and substituting into the difference equation

a.(n+1) + b - (153/160)[a.n + b] = 35

n(a - 153/160) + a + b - (153/160).b = 35

n(a - 153/160) + a + (7/160)b = 35

This is an identity, and so terms in n and the constant can be equated
on both sides of the identity.

Coefficient of n = 0 so   a = 153/160

and equating the constant term on both sides gives

153/160 + (7/160)b = 35

(7/160)b = 5447/160

b = 5447/7

adding v(n) and w(n) we get

u(n) = A.(153/160)^n + (153/160)n + 5447/7

and u(1) = 35 so

35 = (A+1).(153/160) + 5447/7

(153/160)(A+1) = -5202/7

A+1 = -5440/7

A = -5447/7

u(n) = (5447/7)[1 - (153/160)^n] + (153/160)n

and if n = 23 this gives

u(n) = 500.0418 + 21.99375

=  522.035

So after 23 paintings there will be 522 of the 800 balls painted red.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
College Probability
High School Probability

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