Calculating the Probability of Drawing a White Ball from Randomly Chosen Boxes

In this article, we explore the method to calculate the probability of drawing a white ball from one of two randomly chosen boxes. We will use the concept of the law of total probability and basic probability principles to derive the answer. By the end of this article, you will understand how to systematically approach such problems and enhance your knowledge of conditional probability and random selection.

Introduction to the Problem

Consider two boxes, each containing a mix of black and white balls. Box 1 contains 1 black ball and 2 white balls, while Box 2 contains 3 black balls and 5 white balls. A box is chosen at random, and then a ball is drawn from the selected box. We aim to determine the probability that the ball drawn is white.

Step-by-Step Solution

Step 1: Determine the Probabilities of Selecting Each Box

The problem states that a box is chosen randomly. Therefore, the probability of selecting either box is equal.

[ P(text{Box 1}) frac{1}{2} ]

[ P(text{Box 2}) frac{1}{2} ]

Step 2: Determine the Probabilities of Drawing a White Ball from Each Box

For Box 1, we have a total of 3 balls (1 black and 2 white). The probability of drawing a white ball from Box 1 is:

[ P(text{White} | text{Box 1}) frac{2}{3} ]

For Box 2, we have a total of 8 balls (3 black and 5 white). The probability of drawing a white ball from Box 2 is:

[ P(text{White} | text{Box 2}) frac{5}{8} ]

Step 3: Apply the Law of Total Probability

The law of total probability states that for any two events A and B, and their conditional probabilities, the total probability of event A can be found by summing the product of the probability of each event and its conditional probability. Mathematically, it can be represented as:

[ P(A) P(B_1) cdot P(A | B_1) P(B_2) cdot P(A | B_2) ]

Substituting the values we have:

[ P(text{White}) P(text{Box 1}) cdot P(text{White} | text{Box 1}) P(text{Box 2}) cdot P(text{White} | text{Box 2}) ]

Let's perform the calculations step-by-step: For Box 1:

[ P(text{Box 1}) cdot P(text{White} | text{Box 1}) frac{1}{2} cdot frac{2}{3} frac{1}{3} ]

For Box 2:

[ P(text{Box 2}) cdot P(text{White} | text{Box 2}) frac{1}{2} cdot frac{5}{8} frac{5}{16} ]

Step 4: Combine the Results

To find the total probability of drawing a white ball, we need to add the probabilities obtained for each box. First, we need to express (frac{1}{3}) and (frac{5}{16}) with a common denominator, which is 48. (frac{1}{3} frac{16}{48}) (frac{5}{16} frac{15}{48}) Now, we can add these fractions:

[ P(text{White}) frac{16}{48} frac{15}{48} frac{31}{48} ]

Conclusion

The probability that the ball drawn is white is (boxed{frac{31}{48}}).

Further Exploration

To further solidify your understanding, you can explore similar problems involving different numbers of balls and probabilities. Understanding the law of total probability and conditional probability will enable you to tackle more complex probability problems confidently.

Related Keywords

- probability - law of total probability - random selection