Question

A coin is tossed three times. Let A be the event of "getting three heads" and B be the event of "getting a head on the first toss". Then A and B are:

• A. Dependent events
• B. Independent events
• C. Impossible events
• D. Certain events

Hint:

To check if events are independent, calculate the product of their probabilities and compare it with the probability of their intersection.

Answer 1

The Correct Option is B

We are given two events: - Event A: "Getting three heads in three tosses" - Event B: "Getting a head on the first toss" Step 1: Understand the nature of events - Event A occurs only if all three tosses result in heads. So, the probability of A occurring is \( P(A) = \frac{1}{8} \) (since there are 8 possible outcomes from tossing the coin three times, and only one outcome results in three heads). - Event B occurs if the first toss is a head. The probability of B occurring is \( P(B) = \frac{1}{2} \) (since the first toss can either be a head or a tail, and both have equal likelihood). Step 2: Determine if the events are independent Two events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, this is defined as: \[ P(A \cap B) = P(A) \times P(B) \] For this case: - \( P(A \cap B) \) is the probability that both A and B occur, i.e., the first toss is a head, and all three tosses result in heads. This is just \( P(A \cap B) = P(A) = \frac{1}{8} \), because if A occurs, then B automatically occurs. - \( P(A) \times P(B) = \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \) Since \( P(A \cap B) = P(A) \), the events are independent. Thus, the correct answer is option (2), Independent events.