Chapter 16: Problem 3

There are 3 red and 2 white balls in one box and 4 red and 5 white in the second box. You select a box at random and from it pick a ball at random. If the ball is red, what is the probability that it came from the second box?

Short Answer

Expert verified

The probability that the ball came from the second box given that it is red is \( \frac{20}{47} \).

Step by step solution

- Define Events

Let Box 1 be B1 and Box 2 be B2. Let Red Ball be R. We need to find P(B2|R), the probability that the ball came from the second box given that it is red.

- Use Bayes' Theorem

Bayes' theorem states: \[ P(B2|R) = \frac{P(R|B2) \, P(B2)}{P(R)} \] where, P(B1) = P(B2) = 0.5 (since you select a box at random)

- Calculate Probability of Selecting a Red Ball from Each Box

\[ P(R|B1) = \frac{3}{3+2} = \frac{3}{5} \] and \[ P(R|B2) = \frac{4}{4+5} = \frac{4}{9} \]

- Calculate the Total Probability of Selecting a Red Ball

\[ P(R) = P(R|B1) \, P(B1) + P(R|B2) \, P(B2) = \left( \frac{3}{5} \times 0.5 \right) + \left( \frac{4}{9} \times 0.5 \right) = \frac{3}{10} + \frac{2}{9} = \frac{27 + 20}{90} = \frac{47}{90} \]

- Final Calculation

Using Bayes' theorem, we get: \[ P(B2|R) = \frac{\left( \frac{4}{9} \times 0.5 \right)}{\frac{47}{90}} = \frac{2}{9} \times \frac{90}{47} = \frac{20}{47} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

conditional probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. In our problem, we need to find the probability that a red ball came from the second box, knowing that the drawn ball is red. This is symbolized as P(B2|R). With conditional probability, you're focusing on a subset of the entire sample space. For our problem, the subset is the event where the ball is red, and we consider how likely it originated from box 2.

probability theory

Probability theory involves studying how likely events are to happen. It provides a mathematical way to measure uncertainty. Various rules and theorems within this field help to calculate probabilities.
In our problem, we use the axioms of probability to define initial probabilities like P(B1) and P(B2). Each box has equal likelihood, thus P(B1) = P(B2) = 0.5.
We also calculate specific event probabilities, such as the chance of drawing a red ball from each box: P(R|B1) = 3/5 for box 1 and P(R|B2) = 4/9 for box 2.

probabilistic reasoning

Probabilistic reasoning is the method of applying probability to make predictions or reach conclusions. It involves understanding how different probabilities interact and influence one another.
In our problem, it's used to structure the solution logically:

First, we identify key events: drawing from box 1 (B1) and drawing from box 2 (B2).
Next, we find the probability of drawing a red ball from each box.
We then use this information to determine the overall probability of drawing a red ball (P(R)).

This reasoning process culminates in applying Bayes' Theorem to find our desired probability.

bayesian inference

Bayesian inference involves updating our belief about a probability after obtaining new evidence.
We applied Bayes' Theorem to find the probability that the red ball came from the second box, given that it is red. Bayes' Theorem is given by:
P(B2|R) = (P(R|B2) * P(B2)) / P(R).

Applying the values, we get:
P(B2|R) = (4/9 * 0.5) / (47/90) = 20/47.
Bayesian inference thus enables us to update probabilities based on new data. It's a crucial method in probability and statistics.