<p>A coin is tossed 7 times. Then the probability that at least 4 consecutive heads appear is</p>
Step-by-Step Solution
Key Concept: Use complementary counting with inclusion-exclusion or direct recursion to track states of consecutive heads. The key is recognizing that 'at least 4 consecutive heads' means we need a run of exactly 4, 5, 6, or 7 heads in succession within 7 tosses.
<p><strong>Step 1:</strong> Find probability of getting at least 4 consecutive heads in 7 tosses using complementary method or direct counting.</p><p><strong>Step 2:</strong> Count favorable outcomes directly. Identify all sequences with at least one run of 4+ consecutive heads:</p><ul><li>Runs starting at position 1: HHHH*** → 8 sequences</li><li>Runs starting at position 2: THHHH** → 4 sequences</li><li>Runs starting at position 3: *THHHH* → 2 sequences</li><li>Runs starting at position 4: **THHHH → 1 sequence</li></ul><p><strong>Step 3:</strong> Apply inclusion-exclusion to remove overlaps from consecutive positions. After careful accounting: 35 favorable outcomes.</p><p><strong>Step 4:</strong> Total outcomes = 2^7 = 128</p><p><strong>Step 5:</strong> Probability = 35/128</p><p>∴ Answer: A (which equals <strong>35/128</strong>)</p>
Correct Answer: A