Red Marble Probability

Solution & Explanation

To solve the problem of finding the probability that a randomly selected red marble came from Urn #1, we will use Bayes' theorem. This theorem allows us to calculate conditional probabilities, which are the probabilities of an event occurring given that another event has occurred.

Definitions and Probabilities:

• Event A: The marble was pulled from Urn #1.
• Event A': The marble was pulled from Urn #2.
• Event B: The marble is red.

Given Information:

• Urn #1 contains 30 red marbles and 10 black marbles.
• Urn #2 contains 20 red marbles and 20 black marbles.

Total Marbles:

• Total marbles in Urn #1 = 30 + 10 = 40
• Total marbles in Urn #2 = 20 + 20 = 40
• Total marbles overall = 40 + 40 = 80

Probabilities:

• Probability of selecting Urn #1: $P(A) = \frac{40}{80} = \frac{1}{2}$
• Probability of selecting Urn #2: $P(A') = \frac{40}{80} = \frac{1}{2}$

Conditional Probabilities:

• Probability of selecting a red marble from Urn #1: $P(B|A) = \frac{30}{40} = \frac{3}{4}$
• Probability of selecting a red marble from Urn #2: $P(B|A') = \frac{20}{40} = \frac{1}{2}$

Total Probability of Selecting a Red Marble (Event B): Using the law of total probability: $P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A')$ Substitute the values: $P(B) = \left(\frac{3}{4} \times \frac{1}{2}\right) + \left(\frac{1}{2} \times \frac{1}{2}\right)$ $P(B) = \frac{3}{8} + \frac{1}{4} = \frac{3}{8} + \frac{2}{8} = \frac{5}{8}$

Using Bayes' Theorem to Find $P(A|B)$: Bayes' theorem: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$ Substitute the values: $P(A|B) = \frac{\left(\frac{3}{4} \times \frac{1}{2}\right)}{\frac{5}{8}}$ $P(A|B) = \frac{\frac{3}{8}}{\frac{5}{8}} = \frac{3}{5}$

Conclusion: The probability that the red marble was drawn from Urn #1, given that it is red, is $\frac{3}{5}$ or 0.6, which means there is a 60% likelihood that the red marble came from Urn #1.