ITPEC FE Morning April 2020 Question 2

0.8125 — subtract the outcomes with fewer than two heads from all 32 equally likely outcomes.

• •All outcomes: 2⁵ = 32
• •No heads: C(5, 0) = 1
• •Exactly one head: C(5, 1) = 5
• •At least two heads: 32 − 1 − 5 = 26, and 26 ÷ 32 = 0.8125

Check directly: C(5, 2) + C(5, 3) + C(5, 4) + C(5, 5) = 10 + 10 + 5 + 1 = 26.

Answer (b)

Why not others:

• •(a) 0.78125 — 25/32 adds the two-, three- and four-head cases but misses the all-heads outcome
• •(c) 0.84375 — 27/32 subtracts only the five one-head outcomes and forgets the no-heads outcome
• •(d) 0.875 — 28/32 would leave only 4 outcomes with fewer than two heads, but there are 6 (one with none, five with one)

Key rule: For "at least k", subtract the complementary cases from the total when there are fewer of them, and list every one, including the zero case.

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