MHB Why is there a probability paradox in this bag of balls?

AI Thread Summary
The discussion revolves around a probability paradox involving a bag of balls, initially containing two balls that can be either black or white. Participants analyze the probabilities of different combinations of balls after adding a black ball, leading to confusion about the actual contents of the bag. The key point is that the initial assumption of three possible combinations (BB, BW, WW) is incorrect; there are actually four equally likely outcomes when considering the addition of a black ball. This misunderstanding highlights the difference between expectation and actual content, clarifying that the probability of drawing a black ball is consistently calculated as 2/3. Ultimately, the conversation emphasizes the importance of correctly framing probability scenarios to avoid paradoxes.
soroban
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A bag contains two balls.
Either ball can be black or white.

Without drawing any balls, determine the colors of the balls.Solution

The bag could contain any of three contents, each with probabiity \tfrac{1}{3}.

. . \boxed{B B} \qquad \boxed{BW} \qquad \boxed{WW}

Add a black ball to the bag.
The resulting contents are

. . \boxed{BB \,B} \qquad \boxed{BW\,B} \qquad \boxed {WW\,B}Consider the probability of drawing a black ball.

Then: .P(B) \;=\;(\tfrac{1}{3})(\tfrac{3}{3}) + (\tfrac{1}{3})(\tfrac{2}{3}) + (\tfrac{1}{3})(\tfrac{1}{3}) \;=\;\tfrac{6}{9} \;=\;\tfrac{2}{3}Since the probability of drawing a black ball is \tfrac{2}{3}.
. . there must be 2 black and 1 white ball in the bag.

Therefore, before the black ball was added,
. . the bag must have contained one black and one white ball.
 
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My head hurts. (Crying)

-Dan
 
The trick is in your first statement,
soroban said:
The bag could contain any of three contents, each with probabiity \tfrac{1}{3}.
.

There are four equally likely outcomes, WW, WB, BW, and BB. Each has probability 1/4.
 
HallsofIvy said:
The trick is in your first statement,
.

There are four equally likely outcomes, WW, WB, BW, and BB. Each has probability 1/4.
his makes no difference.
We still add one black ball.

\text{We have: }\;WW\!B,\;W\!BB,\;BW\!B,\;BBB

\text{Then we have:}
. . \begin{array}{cccccc}<br /> P(B|WW\!B) &amp;=&amp; \frac{1}{4}\cdot \frac{1}{3} &amp;=&amp; \frac{1}{12} \\<br /> P(B|W\!BB) &amp;=&amp; \frac{1}{4} \cdot \frac{2}{3} &amp;=&amp; \frac{1}{6} \\<br /> P(B|BW\!B) &amp;=&amp; \frac{1}{4} \cdot \frac{2}{3} &amp;=&amp; \frac{1}{6} \\<br /> P(B|BBB) &amp;=&amp; \frac{1}{4} \cdot \frac{3}{3} &amp;=&amp; \frac{1}{4} \end{array}

\text{Therefore: }\;P(B) \;=\;\tfrac{1}{12} + \tfrac{1}{6}+\tfrac{1}{6} + \tfrac{1}{4} \;=\;\frac{2}{3}

THe "proof" still holds . . . LOL!
 
Nice conundrum! :)

I believe we've found the expectation, or average if you will, of the number of white and black balls.
 

Hello, ILS!

You've discovered the truth behind this apparent paradox.
This is indeed a matter of expectation, not actual content.

Consider the original bag with two balls
The bag could contain any of three contents, each with probability \tfrac{1}{3}.
. . \boxed{BB}\quad \boxed{BW} \quad \boxed{WW}

Consider the probability of drawing a black ball.,

P(B) \;=\;\left(\tfrac{1}{3}\right)\left(\tfrac{2}{2}\right) + \left(\tfrac{1}{3}\right)\left(\tfrac{1}{2}\right) + \left(\tfrac{1}{3}\right)\left(\tfrac{0}{2}\right) \;=\; \frac{1}{2}

We would NOT conclude that the bag must contain one black and one white ball.
But that is exactly what we did with the bag with three balls.
 
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