<p>A fair coin is tossed <em>n</em> times. If the probability that head occurs 6 times is equal to the probability that head occurs 8 times, then find the value of <em>n</em>.</p>
Step-by-Step Solution
Key Concept: Use the binomial probability formula P(X=k) = C(n,k)·p^k·(1-p)^(n-k). Since P(H=6) = P(H=8) and p=1/2, the binomial coefficients must be equal: C(n,6) = C(n,8), which occurs only when n-6 = 8 (using the symmetry property C(n,r) = C(n,n-r)).
<p><strong>Step 1:</strong> Write the probability formula for a fair coin tossed n times:</p><p>P(H=6) = C(n,6)·(1/2)^6·(1/2)^(n-6) = C(n,6)·(1/2)^n</p><p>P(H=8) = C(n,8)·(1/2)^8·(1/2)^(n-8) = C(n,8)·(1/2)^n</p><p><strong>Step 2:</strong> Set P(H=6) = P(H=8):</p><p>C(n,6)·(1/2)^n = C(n,8)·(1/2)^n</p><p>⟹ C(n,6) = C(n,8)</p><p><strong>Step 3:</strong> Apply the symmetry property C(n,r) = C(n,n-r):</p><p>C(n,6) = C(n,8) means either 6 = 8 (impossible) or 6 = n-8</p><p><strong>Step 4:</strong> Solve for n:</p><p>6 = n - 8</p><p>n = 14</p><p><strong>Verification:</strong> C(14,6) = C(14,8) ✓ (both equal 3003)</p><p>∴ Answer: 14</p>
Correct Answer: 14