Probability
Classical Probability
Grade 12
Question:
<p><strong>For Problems 13–15:</strong> An amoeba either splits into two or remains the same or eventually dies out immediately after completion of every second with probabilities, respectively, 1/2, 1/4, and 1/4. Let the initial amoeba be called as mother amoeba and after every second, the amoeba, if it is distinct from the previous one, be called as 2nd, 3rd, ... generations.</p><p>The probability that after 2 s exactly 4 amoeba are alive is</p>
<p>(1) 1/16</p>
<p>(2) 1/8</p>
<p>(3) 1/4</p>
<p>(4) 1/2</p>
Step-by-Step Solution
Key Concept: Track the population transitions through each second by identifying all possible paths that lead to exactly 4 amoebas after 2 seconds. After 1 second, we need 2 amoebas (each must split), then after 2 seconds, each of those 2 must split to give 4 total.
<p><strong>Step 1:</strong> For exactly 4 amoebas after 2 seconds, trace the necessary population path.</p><p><strong>Step 2:</strong> At t = 1s: Mother amoeba must split into 2 amoebas (probability = 1/2). Any other outcome (dies or stays same) cannot lead to 4 amoebas.</p><p><strong>Step 3:</strong> At t = 2s: Both of the 2 amoebas from generation 1 must independently split into 2 each (probability = 1/2 for each). This gives 2 × 2 = 4 amoebas.</p><p><strong>Step 4:</strong> Calculate total probability: P(4 amoebas at t=2s) = P(mother splits) × P(1st offspring splits) × P(2nd offspring splits) = (1/2) × (1/2) × (1/2) = 1/8</p><p>∴ Answer: A</p>
Correct Answer: A