Problem 18 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

One bag contains 5 white and 4 black balls. Another bag contains 7 white and 9 black balls. A ball is transferred from the first bag to the second bag and then a ball is drawn from the second bag. The probability that the ball is white is (a) \(\frac{8}{17}\) (b) \(\frac{40}{153}\) (c) \(\frac{5}{9}\) (d) \(\frac{4}{9}\)

Step-by-Step Solution

Answer

The probability that the drawn ball is white is \( \frac{40}{153} \).

Step 1: Define the Initial Conditions

Bag 1 contains 5 white balls and 4 black balls, so 9 balls in total. Bag 2 contains 7 white balls and 9 black balls, totaling 16 balls. We'll calculate probabilities based on these initial numbers.

Step 2: Transfer a White Ball from Bag 1 to Bag 2

The probability of transferring a white ball from Bag 1 is \( \frac{5}{9} \). If a white ball is transferred, Bag 2 will then have 8 white balls and 9 black balls, totaling 17 balls.

Step 3: Transfer a Black Ball from Bag 1 to Bag 2

The probability of transferring a black ball from Bag 1 is \( \frac{4}{9} \). If a black ball is transferred, Bag 2 will have 7 white balls and 10 black balls, totaling 17 balls.

Step 4: Calculate Probability of Drawing a White Ball after Transferring a White Ball

If a white ball is transferred (probability \( \frac{5}{9} \)), the probability of drawing a white ball from Bag 2 is \( \frac{8}{17} \).

Step 5: Calculate Probability of Drawing a White Ball after Transferring a Black Ball

If a black ball is transferred (probability \( \frac{4}{9} \)), the probability of drawing a white ball from Bag 2 is \( \frac{7}{17} \).

Step 6: Calculate Total Probability of Drawing a White Ball

Using the law of total probability, the overall probability of drawing a white ball from Bag 2 is the sum of two joint probabilities: \[ \left( \frac{5}{9} \times \frac{8}{17} \right) + \left( \frac{4}{9} \times \frac{7}{17} \right) = \frac{40}{153} \].

Key Concepts

Conditional Probability

Imagine having two events, where the occurrence of the first event impacts the probability of the second event. This scenario is the core of Conditional Probability. In simpler terms, conditional probability tests how the probability of one event is affected or changed by the occurrence of another event.

To calculate the conditional probability, use the formula: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\]Here, \(P(A|B)\) is the probability of event A occurring given that B has occurred, \(P(A \cap B)\) is the probability that both events occur, and \(P(B)\) is the probability of B.

In the context of the exercise, transferring a ball affects the probability of drawing a white ball from the second bag. For instance, once a white ball is transferred from Bag 1 to Bag 2, the number of white balls in Bag 2 changes, affecting the conditions for drawing a white ball.

Total Probability

The Total Probability Rule helps in calculating the probability of an event by considering all possible scenarios through which the event can occur. It's particularly useful when dealing with multiple probabilities happening in sequence or parallel.

For this exercise, the law of total probability adds up the probabilities of drawing a white ball after considering both scenarios (transferring a white or black ball). The formula used is:\[P(\text{Draw White}) = P(\text{White Transfer}) \cdot P(\text{Draw White|White Transfer}) + P(\text{Black Transfer}) \cdot P(\text{Draw White|Black Transfer})\]This formula ensures all avenues of the problem are addressed, leading to a comprehensive total probability. It's a methodical approach ensuring no possibility is left unchecked.

Probability Distribution

Probability distribution describes how the probabilities are spread over various possible outcomes for a random variable. It gives us a complete picture of where chances lie in any given experiment or scenario.

Consider the bags of balls as a small probability distribution snapshot. When a ball is transferred from Bag 1 to Bag 2, the count of balls (white and black) changes, impacting the distribution of probability for drawing any specific type of ball.

• Bag 1 starts with a clear division: 5 whites, 4 blacks.
• Bag 2's initial distribution: 7 whites, 9 blacks.

With each possible transfer from Bag 1 to Bag 2, these numbers adjust, thus shifting the probability distribution of drawing from the second bag. This exemplifies how a probability distribution modifies with changes in initial conditions or transfers, hugely affecting outcomes.