Probability
Classical Probability
Grade 12
Question:
<p>If ten objects are distributed at random among ten persons, find the probability that at least one of them will not get any object.</p>
<p>\(\dfrac{10^{10} - 10!}{10^{10}}\)</p>
<p>\(\dfrac{10!}{10^{10}}\)</p>
<p>\(\dfrac{10^{10} + 10!}{10^{10}}\)</p>
<p>\(1 - \dfrac{1}{10!}\)</p>
Step-by-Step Solution
Key Concept: Use complementary counting: P(at least one person gets nothing) = 1 - P(all persons get at least one object). The latter equals the probability that the distribution is a surjection, which requires exactly one object per person, giving probability 10!/10^10.
<p><strong>Step 1:</strong> Find the complementary event. At least one person gets nothing = complement of all 10 persons getting at least one object.</p><p><strong>Step 2:</strong> Total ways to distribute 10 objects among 10 persons = 10^10 (each object can go to any of 10 persons).</p><p><strong>Step 3:</strong> Ways that all 10 persons get at least one object (surjections) = 10! (a bijection or one-to-one correspondence, where each person gets exactly one object). By inclusion-exclusion: Surjections = Σ(-1)^k · C(10,k) · (10-k)^10 for k=0 to 9, which equals 10! when computed correctly.</p><p><strong>Step 4:</strong> Probability all get at least one = 10!/10^10</p><p><strong>Step 5:</strong> P(at least one gets nothing) = 1 - 10!/10^10 = (10^10 - 10!)/10^10</p><p>∴ Answer: A</p>
Correct Answer: A