Probability
Total Probability Theorem
Grade 12
Question:
<p>A bag contains <em>n</em> + 1 coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is 7/12, then find the value of <em>n</em>.</p>
Step-by-Step Solution
Key Concept: Use the law of total probability by conditioning on whether the selected coin is the two-headed coin or a fair coin. The total probability of heads equals the sum of probabilities when each type of coin is selected.
<p><strong>Step 1:</strong> Set up the problem using total probability.</p><p>Total coins = n + 1 (where 1 is two-headed, and n are fair coins)</p><p>P(selecting two-headed coin) = 1/(n+1)</p><p>P(selecting a fair coin) = n/(n+1)</p><p><strong>Step 2:</strong> Apply the law of total probability for heads.</p><p>P(Heads) = P(Heads | two-headed) · P(two-headed) + P(Heads | fair) · P(fair)</p><p>7/12 = 1 · [1/(n+1)] + (1/2) · [n/(n+1)]</p><p><strong>Step 3:</strong> Solve for n.</p><p>7/12 = 1/(n+1) + n/[2(n+1)]</p><p>7/12 = [2 + n]/[2(n+1)]</p><p>7 · 2(n+1) = 12(2 + n)</p><p>14n + 14 = 24 + 12n</p><p>2n = 10</p><p>∴ n = 5</p>
Correct Answer: 5