There are 5 urns numbered 1 to 5. Each urn contains 10 balls. The ith urn has i defective balls and 10 - i non-defective balls (i = 1, 2, ..., 5). An urn is chosen at random and th... There are 5 urns numbered 1 to 5. Each urn contains 10 balls. The ith urn has i defective balls and 10 - i non-defective balls (i = 1, 2, ..., 5). An urn is chosen at random and then a ball is selected from that urn. a) What is the probability that a defective ball is selected? b) If the selected ball is defective, find the probability that it came from urn i (i = 1, 2, ..., 5). Read more

Steps to Solve

1. Calculate the probability of selecting a defective ball from each urn

Each urn has a different number of defective balls:

• Urn 1: 1 defectives, 9 non-defectives
• Urn 2: 2 defectives, 8 non-defectives
• Urn 3: 3 defectives, 7 non-defectives
• Urn 4: 4 defectives, 6 non-defectives
• Urn 5: 5 defectives, 5 non-defectives

The probability of selecting a defective ball from each urn (i) is given by:

$$ P(D|U_i) = \frac{i}{10} $$

2. Find the total probability of selecting a defective ball

Since each urn is equally likely to be chosen, we will use the law of total probability:

$$ P(D) = \sum_{i=1}^{5} P(U_i) P(D|U_i) $$

Each urn has a probability of ( P(U_i) = \frac{1}{5} ):

$$ P(D) = \frac{1}{5}\left(\frac{1}{10} + \frac{2}{10} + \frac{3}{10} + \frac{4}{10} + \frac{5}{10}\right) $$

Calculating this gives:

$$ P(D) = \frac{1}{5} \cdot \frac{15}{10} = \frac{15}{50} = \frac{3}{10} $$

3. Calculate the conditional probability each urn given defective

Using Bayes' theorem, we find the probability that a defective ball came from urn (i):

$$ P(U_i|D) = \frac{P(D|U_i)P(U_i)}{P(D)} $$

Each urn has:

• Urn 1:

$$ P(U_1|D) = \frac{\frac{1}{10} \cdot \frac{1}{5}}{\frac{3}{10}} = \frac{1}{6} $$

• Urn 2:

$$ P(U_2|D) = \frac{\frac{2}{10} \cdot \frac{1}{5}}{\frac{3}{10}} = \frac{1}{3} $$

• Urn 3:

$$ P(U_3|D) = \frac{\frac{3}{10} \cdot \frac{1}{5}}{\frac{3}{10}} = \frac{1}{5} $$

• Urn 4:

$$ P(U_4|D) = \frac{\frac{4}{10} \cdot \frac{1}{5}}{\frac{3}{10}} = \frac{2}{5} $$

• Urn 5:

$$ P(U_5|D) = \frac{\frac{5}{10} \cdot \frac{1}{5}}{\frac{3}{10}} = \frac{1}{3} $$

4. Summarize results for urns

For each urn (i):

• ( P(U_1|D) = \frac{1}{6} )
• ( P(U_2|D) = \frac{1}{3} )
• ( P(U_3|D) = \frac{1}{5} )
• ( P(U_4|D) = \frac{2}{5} )
• ( P(U_5|D) = \frac{1}{3} )