Question

A die is thrown twice. Let A be the event of getting a prime number when the die is thrown first time and B be the event of getting an even number when the die is thrown second time. Then \(P(A/B) =\)

• A. \(\frac{1}{2} \)
• B. \(\frac{2}{3} \)
• C. \(\frac{1}{5} \)
• D. \(\frac{3}{5} \)

Hint:

Recognizing whether events are independent is crucial in probability. If two events A and B are independent, then the conditional probability \(P(A|B)\) is equal to \(P(A)\). This is because the occurrence of B provides no new information that changes the likelihood of A.

Answer 1

The Correct Option is A

Step 1: Identify the possible outcomes for a single die throw.
When a standard six-sided die is thrown, the set of all possible outcomes is \(\{1, 2, 3, 4, 5, 6\}\). The total number of outcomes is 6. 
Step 2: Define Event A and calculate its probability.
Event A: Getting a prime number when the die is thrown the first time. Prime numbers in the set \(\{1, 2, 3, 4, 5, 6\}\) are \(\{2, 3, 5\}\). The number of outcomes favorable to A is 3. The probability of event A is: \[ P(A) = \frac{\text{Number of prime numbers}}{\text{Total number of outcomes}} = \frac{3}{6} = \frac{1}{2} \] Step 3: Define Event B and calculate its probability.
Event B: Getting an even number when the die is thrown the second time.
Even numbers in the set \(\{1, 2, 3, 4, 5, 6\}\) are \(\{2, 4, 6\}\).
The number of outcomes favorable to B is 3.
The probability of event B is:
\[ P(B) = \frac{\text{Number of even numbers}}{\text{Total number of outcomes}} = \frac{3}{6} = \frac{1}{2} \] Step 4: Determine the relationship between Event A and Event B.
The first die throw and the second die throw are independent events. The outcome of the first throw does not affect the outcome of the second throw, and vice versa. 
Step 5: Calculate the conditional probability \(P(A/B)\).
For independent events, the conditional probability of A given B is simply the probability of A. \[ P(A/B) = P(A) \] Therefore, \[ P(A/B) = \frac{1}{2} \] The final answer is $\boxed{\frac{1}{2}}$.