Question

Two fair dice, one having red and another having blue color, are tossed independently once. Let \( A \) be the event that the die having red colour will show 5 or 6. Let \( B \) be the event that the sum of the outcomes will be 7 and let \( C \) be the event that the sum of the outcomes will be 8. Then which one of the following statements is true?

• A. \( A \) and \( B \) are independent as well as \( A \) and \( C \) are independent
• B. \( A \) and \( B \) are independent, but \( A \) and \( C \) are not independent
• C. \( A \) and \( C \) are independent, but \( A \) and \( B \) are not independent
• D. Neither \( A \) and \( B \) are independent, nor \( A \) and \( C \) are independent

Hint:

Two events are independent if and only if \( P(A \cap B) = P(A) \cdot P(B) \).

Answer 1

The Correct Option is B

To solve this problem, we need to determine the independence of the events \( A \), \( B \), and \( C \). Let's analyze each event and their independence step by step.

1. Calculate the probability of each event:

- Event \( A \) is described as "the die having red color shows 5 or 6". Each face on a dice has an equal probability of landing on top, so:
\(P(A) = \frac{2}{6} = \frac{1}{3}\)

- Event \( B \) is "the sum of the outcomes is 7". The possible favorable outcomes are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), leading to:
\(P(B) = \frac{6}{36} = \frac{1}{6}\)

- Event \( C \) is "the sum of the outcomes is 8". The favorable outcomes are (2,6), (3,5), (4,4), (5,3), and (6,2), resulting in:
\(P(C) = \frac{5}{36}\)

1. Calculate \( P(A \cap B) \) and \( P(A \cap C) \):

- For \( A \cap B \), the red die can be 5 or 6, so the favorable outcomes are (5,2), (6,1) leading to:
\(P(A \cap B) = \frac{2}{36} = \frac{1}{18}\)

- For \( A \cap C \), the favorable outcomes are the same possibilities that give a sum of 8 involving 5 or 6 on a red die, such as (5,3), (6,2):
\(P(A \cap C) = \frac{2}{36} = \frac{1}{18}\)

1. Check independence of events using the formula for independent events: \(P(X \cap Y) = P(X) \times P(Y)\).

- For \( A \) and \( B \):
\(P(A) \times P(B) = \frac{1}{3} \times \frac{1}{6} = \frac{1}{18}\)
Since \(P(A \cap B) = \frac{1}{18}\), \( A \) and \( B \) are independent.

- For \( A \) and \( C \):
\(P(A) \times P(C) = \frac{1}{3} \times \frac{5}{36} = \frac{5}{108}\)
Since \(\frac{5}{108} \ne \frac{1}{18}\), \( A \) and \( C \) are not independent.

1. Conclusion:

The correct option is "\((A) \text{ and } (B) \text{ are independent, but } (A) \text{ and } (C) \text{ are not independent}\)".

Answer 2

Step 1: Understand the events.
- Event \( A \) represents the outcome where the red die shows 5 or 6. Hence, the probability of \( A \) is: \[ P(A) = P(\text{Red die shows 5 or 6}) = \frac{2}{6} = \frac{1}{3} \] - Event \( B \) represents the event that the sum of the outcomes of the two dice is 7. The possible pairs for a sum of 7 are: \[ (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \] There are 6 outcomes that result in a sum of 7 out of a total of \( 6 \times 6 = 36 \) possible outcomes. Therefore: \[ P(B) = \frac{6}{36} = \frac{1}{6} \] - Event \( C \) represents the event that the sum of the outcomes of the two dice is 8. The possible pairs for a sum of 8 are: \[ (2,6), (3,5), (4,4), (5,3), (6,2) \] There are 5 outcomes that result in a sum of 8 out of 36 total possible outcomes. Therefore: \[ P(C) = \frac{5}{36} \] Step 2: Check if events \( A \) and \( B \) are independent.
To check if two events are independent, we verify if the following holds: \[ P(A \cap B) = P(A) \cdot P(B) \] - \( A \cap B \) is the event where the red die shows 5 or 6 and the sum of the dice is 7. The pairs that satisfy both conditions are: \[ (5,2), (6,1) \] Thus, there are 2 outcomes that satisfy both \( A \) and \( B \), so: \[ P(A \cap B) = \frac{2}{36} = \frac{1}{18} \] Now, check if \( P(A) \cdot P(B) \) equals \( P(A \cap B) \): \[ P(A) \cdot P(B) = \frac{1}{3} \times \frac{1}{6} = \frac{1}{18} \] Since \( P(A \cap B) = P(A) \cdot P(B) \), events \( A \) and \( B \) are independent.
Step 3: Check if events \( A \) and \( C \) are independent.
Now, check if \( P(A \cap C) = P(A) \cdot P(C) \). - \( A \cap C \) is the event where the red die shows 5 or 6 and the sum of the dice is 8. The pairs that satisfy both conditions are: \[ (5,3), (6,2) \] Thus, there are 2 outcomes that satisfy both \( A \) and \( C \), so: \[ P(A \cap C) = \frac{2}{36} = \frac{1}{18} \] Now, check if \( P(A) \cdot P(C) \) equals \( P(A \cap C) \): \[ P(A) \cdot P(C) = \frac{1}{3} \times \frac{5}{36} = \frac{5}{108} \] Since \( P(A \cap C) = \frac{1}{18} \neq \frac{5}{108} \), events \( A \) and \( C \) are not independent.
Step 4: Conclusion.
The correct statement is (B): \( A \) and \( B \) are independent, but \( A \) and \( C \) are not independent.