Question

An unbiased coin is tossed twice. Let event \( A \) represent getting a head on the first toss, and event \( B \) represent getting a head on the second toss. Determine the mathematical relationship between events \( A \) and \( B \).

• A. Dependent Events
• B. Independent Events
• C. Mutually Exclusive Events
• D. Equivalence Events

Hint:

Do not confuse independent events with mutually exclusive events. Independent events can happen at the same time (\( P(A \cap B) \neq 0 \)), whereas mutually exclusive events can never occur together (\( P(A \cap B) = 0 \)).

Answer 1

The Correct Option is B

Concept: Two events \( A \) and \( B \) are mathematically classified as Independent Events if and only if the occurrence of one does not affect the probability of the other. This condition can be verified using the multiplication rule: \[ P(A \cap B) = P(A) \cdot P(B) \]

Step 1: Define the sample space and calculate individual event probabilities.
Tossing an unbiased coin twice produces a sample space containing 4 equally likely outcomes: \[ S = \{\text{HH}, \, \text{HT}, \, \text{TH}, \, \text{TT}\} \quad \Rightarrow \quad n(S) = 4 \] Map the outcomes for each event and calculate their probabilities:

• Event \( A \) (Head on 1st toss) = \(\{\text{HH}, \text{HT}\} \Rightarrow P(A) = \frac{2}{4} = 0.5\)
• Event \( B \) (Head on 2nd toss) = \(\{\text{HH}, \text{TH}\} \Rightarrow P(B) = \frac{2}{4} = 0.5\)

Step 2: Calculate the joint intersection probability of both events.
The intersection event \( A \cap B \) represents getting a head on both the first and second tosses: \[ A \cap B = \{\text{HH}\} \quad \Rightarrow \quad P(A \cap B) = \frac{1}{4} = 0.25 \]

Step 3: Verify the independent multiplication identity.
Multiply the individual event probabilities together: \[ P(A) \cdot P(B) = 0.5 \times 0.5 = 0.25 \] Since \( P(A \cap B) = P(A) \cdot P(B) = 0.25 \), the two events are independent.