MHB Occupancy Problem (Wackerly & Mendenhall & Schaeffer Problem 2.93)

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The discussion focuses on solving a probability problem involving five bowls, each containing a different number of white and black balls. The first part of the problem requires calculating the probability that both balls selected from a randomly chosen bowl are white, which is found to be 2/5. The second part seeks the probability that bowl 3 was selected given that both balls are white, leading to a corrected answer of 3/20 after reevaluating the calculations. Participants confirm that the reasoning and numerical answers align with the expected results, indicating correctness in the final conclusions. The thread highlights the importance of accurately applying the law of total probability in such scenarios.
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Problem 2.93. Five identical bowls are labeled $1, 2, 3, 4,$ and $5$. Bowl $i$ contains $i$ white and $5-i$ black balls, with $i=1, 2, \dots, 5$. A bowl is randomly selected and two balls are randomly selected (without replacement) from the contents of the bowl.
  • What is the probability that both balls selected are white?
  • Given that both balls selected are white, what is the probability that bowl $3$ was selected?

Answer. Let $B_i$ be the event that bowl $i$ is selected. Let $W_1$ be the event that the first ball taken is white, and let $W_2$ be the event that the second ball taken is white. We are given the following data:
$$P(B_i)=1/5;\quad P(W_1|B_i)=i/5;\quad P(W_2|B_i\cap W_1)=(i-1)/4; \quad P(W_2|B_i\cap \overline{W_1})=i/4.$$
  • We are asked to compute $P(W_2|W_1)$. If $W_1$ has occurred, then the probability of getting a second white ball from bowl $i$ is $(i-1)/4$. So, we find $P(W_2|W_1)$ as follows:
    \begin{align*}
    P(W_2|W_1)&=\sum_{i=1}^{5}P(B_i\cap W_1) \, P(W_2|B_i\cap W_1) \\
    &=\sum_{i=1}^{5}P(B_i) \, P(W_1|B_i) \, P(W_2|B_i\cap W_1) \\
    &=\sum_{i=1}^{5}\left(\frac15 \right) \left(\frac{i}{5} \right)\left(\frac{i-1}{4}\right) \\
    &=\frac{1}{100}\sum_{i=1}^{5}(i^2-i) \\
    &=\frac{1}{100}\left(\frac{5(6)(11)}{6}-15\right) \\
    &=\frac25.
    \end{align*}
  • We are asked to compute $P(B_3|W_1\cap W_2)$. Note that
    $$P(W_1\cap W_2)\, P(B_3|W_1\cap W_2)=P(B_3) \, P(W_1\cap W_2|B_3) $$
    $$\implies P(B_3|W_1\cap W_2)=\frac{P(B_3) \, P(W_1\cap W_2|B_3)}{P(W_1\cap W_2)}
    =\frac{P(B_3) \, P(W_1\cap W_2|B_3)}{P(W_1) \, P(W_2|W_1)}.$$
    To assemble the ingredients of this formula, we note immediately that
    \begin{align*}
    P(B_3)&=\frac15 \\
    P(W_1\cap W_2|B_3)&=\frac35 \cdot \frac24 =\frac35 \cdot \frac12 = \frac{3}{10} \\
    P(W_2|W_1)&=\frac25 \\
    P(W_1)&=\sum_{i=1}^{5}P(B_i) \, P(W_1|B_i)=\frac15 \sum_{i=1}^{5}\frac{i}{5}=
    \frac{15}{25}=\frac35.
    \end{align*}
    It follows, then, that
    $$P(B_3|W_1\cap W_2)=\frac{\frac15 \cdot \frac{3}{10}}{\frac35 \cdot \frac25}
    =\frac{3}{50} \cdot \frac{25}{6}=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}.$$

The first part a is, I am fairly sure, correct. However, I think part b is incorrect. What am I doing wrong? Thank you for your time!
 
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Ackbach said:
Problem 2.93. Five identical bowls are labeled $1, 2, 3, 4,$ and $5$. Bowl $i$ contains $i$ white and $5-i$ black balls, with $i=1, 2, \dots, 5$. A bowl is randomly selected and two balls are randomly selected (without replacement) from the contents of the bowl.
  • What is the probability that both balls selected are white?
  • Given that both balls selected are white, what is the probability that bowl $3$ was selected?

Answer. Let $B_i$ be the event that bowl $i$ is selected. Let $W_1$ be the event that the first ball taken is white, and let $W_2$ be the event that the second ball taken is white. We are given the following data:
$$P(B_i)=1/5;\quad P(W_1|B_i)=i/5;\quad P(W_2|B_i\cap W_1)=(i-1)/4; \quad P(W_2|B_i\cap \overline{W_1})=i/4.$$
  • We are asked to compute $P(W_2|W_1)$. If $W_1$ has occurred, then the probability of getting a second white ball from bowl $i$ is $(i-1)/4$. So, we find $P(W_2|W_1)$ as follows:
    \begin{align*}
    P(W_2|W_1)&=\sum_{i=1}^{5}P(B_i\cap W_1) \, P(W_2|B_i\cap W_1) \\
    &=\sum_{i=1}^{5}P(B_i) \, P(W_1|B_i) \, P(W_2|B_i\cap W_1) \\
    &=\sum_{i=1}^{5}\left(\frac15 \right) \left(\frac{i}{5} \right)\left(\frac{i-1}{4}\right) \\
    &=\frac{1}{100}\sum_{i=1}^{5}(i^2-i) \\
    &=\frac{1}{100}\left(\frac{5(6)(11)}{6}-15\right) \\
    &=\frac25.
    \end{align*}
  • We are asked to compute $P(B_3|W_1\cap W_2)$. Note that
    $$P(W_1\cap W_2)\, P(B_3|W_1\cap W_2)=P(B_3) \, P(W_1\cap W_2|B_3) $$
    $$\implies P(B_3|W_1\cap W_2)=\frac{P(B_3) \, P(W_1\cap W_2|B_3)}{P(W_1\cap W_2)}
    =\frac{P(B_3) \, P(W_1\cap W_2|B_3)}{P(W_1) \, P(W_2|W_1)}.$$
    To assemble the ingredients of this formula, we note immediately that
    \begin{align*}
    P(B_3)&=\frac15 \\
    P(W_1\cap W_2|B_3)&=\frac35 \cdot \frac24 =\frac35 \cdot \frac12 = \frac{3}{10} \\
    P(W_2|W_1)&=\frac25 \\
    P(W_1)&=\sum_{i=1}^{5}P(B_i) \, P(W_1|B_i)=\frac15 \sum_{i=1}^{5}\frac{i}{5}=
    \frac{15}{25}=\frac35.
    \end{align*}
    It follows, then, that
    $$P(B_3|W_1\cap W_2)=\frac{\frac15 \cdot \frac{3}{10}}{\frac35 \cdot \frac25}
    =\frac{3}{50} \cdot \frac{25}{6}=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}.$$

The first part a is, I am fairly sure, correct. However, I think part b is incorrect. What am I doing wrong? Thank you for your time!

Shouldn't the numerator be $P(B_{3}{\cap}W_{1}{\cap}W_{2})=\frac{1}{5}.\frac{3}{5}.\frac{1}{2}=\frac{3}{50}$? EDIT: sorry, that's what you've got. Why do you think the answer's wrong?
 
I've found your problem. In a) you should be finding $P(W_{1}{\cap}W_{2})$, not $P(W_{2}|W_{1})$. Then for b) you get $\frac{3}{50}.\frac{5}{2}=\frac{3}{20}$
 
Ok, so let's do this:

  • We are asked to compute $P(W_1\cap W_2)$. Now the entire sample space
    $\displaystyle S=\bigcup_{i=1}^{5}B_i$, with $B_i\cap B_j=\varnothing$ whenever $i\not=j$. That is, the $B_{i}$ partition $S$. It follows from the Law of Total Probability that
    $$P(W_1\cap W_2)=\sum_{i=1}^{5}P(B_i) \, P(W_1\cap W_2|B_i).$$
    Now
    $$P(W_1\cap W_2|B_i)=\frac{i}{5} \cdot \frac{i-1}{4}=\frac{i^2-i}{20}.$$
    Hence,
    $$P(W_1\cap W_2)=\frac{1}{100}\sum_{i=1}^{5}(i^2-i)=\frac{2}{5}.$$
  • Then, following your lead, we have
    $$P(B_3|W_1\cap W_2)=\frac{P(B_3) \, P(W_1\cap W_2|B_3)}{P(W_1\cap W_2)}
    =\frac{(1/5)(3/5)(2/4)}{2/5}=\frac{3}{20}.$$

Is this reasoning correct? The numerical answers agree with the back of the book, and it certainly seems correct to me.
 
Ackbach said:
Ok, so let's do this:

  • Is this reasoning correct? The numerical answers agree with the back of the book, and it certainly seems correct to me.


  • Yes.
 
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