<p>A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of eleven steps he is one step away from the starting point.</p>
Step-by-Step Solution
Key Concept: The man must take either 6 forward and 5 backward steps (net +1) or 5 forward and 6 backward steps (net -1) to end up exactly one step away. Use binomial probability to find both cases and add them.
<p><strong>Step 1:</strong> Define the problem. After 11 steps, to be exactly 1 step away from start, net displacement = ±1.</p><p><strong>Step 2:</strong> Case 1 - Net displacement = +1: Need 6 forward steps and 5 backward steps.</p><p>P(Case 1) = C(11,6) × (0.4)^6 × (0.6)^5 = 462 × (0.4)^6 × (0.6)^5</p><p><strong>Step 3:</strong> Case 2 - Net displacement = -1: Need 5 forward steps and 6 backward steps.</p><p>P(Case 2) = C(11,5) × (0.4)^5 × (0.6)^6 = 462 × (0.4)^5 × (0.6)^6</p><p><strong>Step 4:</strong> Total probability = P(Case 1) + P(Case 2)</p><p>= 462 × (0.4)^5 × (0.6)^5 × [0.4 + 0.6]</p><p>= 462 × (0.4)^5 × (0.6)^5 × 1</p><p>= 462 × (0.24)^5</p><p>= 462 × (0.0007962624)</p><p><strong>Numerical Calculation:</strong> (0.4)^5 = 0.01024, (0.6)^5 = 0.07776</p><p>= 462 × 0.01024 × 0.07776 ≈ <strong>0.3679</strong> or <strong>C(11,6) × (0.4)^6 × (0.6)^5 + C(11,5) × (0.4)^5 × (0.6)^6</strong></p><p>∴ Answer: <strong>462 × (0.24)^5 ≈ 0.3679 or 36.79%</strong></p>
Correct Answer: 462