<p>A coin is tossed three times. Event <i>A</i>: two heads appear. Event <i>B</i>: last should be head. Then identify whether events <i>A</i> and <i>B</i> are independent or not.</p>
<p>A and B are independent</p>
<p>A and B are not independent</p>
<p>Cannot be determined</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Two events are independent if P(A∩B) = P(A)·P(B). Calculate the probability of both events occurring together and compare with the product of individual probabilities.
<p><strong>Step 1: Find P(A) - probability of exactly 2 heads in 3 tosses</strong></p><p>Favorable outcomes: HHT, HTH, THH → 3 outcomes</p><p>P(A) = 3/8</p><p><strong>Step 2: Find P(B) - probability that last toss is head</strong></p><p>Favorable outcomes: HHH, HTH, THH, TTH → 4 outcomes</p><p>P(B) = 4/8 = 1/2</p><p><strong>Step 3: Find P(A∩B) - probability of exactly 2 heads AND last is head</strong></p><p>This means: exactly 2 heads total with last one being head</p><p>So first two tosses have exactly 1 head: HTH, THH → 2 outcomes</p><p>P(A∩B) = 2/8 = 1/4</p><p><strong>Step 4: Check independence condition</strong></p><p>P(A)·P(B) = (3/8)·(1/2) = 3/16</p><p>P(A∩B) = 2/8 = 4/16</p><p>Since 3/16 ≠ 4/16, we have P(A∩B) ≠ P(A)·P(B)</p><p>∴ <strong>Events A and B are NOT independent (dependent events)</strong></p>
Correct Answer: A