Probability
Derangements, probability of at least k correct
nta_pyq_2023_jan
Grade 12
Question:
Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is
Step-by-Step Solution
Key Concept: Use inclusion-exclusion: P(at least 3 correct) = 1 - P(0 correct) - P(1 correct) - P(2 correct); use derangement numbers $D_n \approx n!/e$
P(at least 3 correct) $= 1 - \frac{D_{15}+\binom{15}{1}D_{14}+\binom{15}{2}D_{13}}{15!}$. Using $D_n \approx n!/e$: $= 1 - \frac{15!/e + 15 \cdot 14!/e + 105 \cdot 13!/e}{15!} = 1-(1/e+1/e+1/(2e)) = 1-5/(2e) \approx 1/6$. Answer: (3)
Correct Answer: $\frac{1}{6}$