Probability
Probability
Allen Star Batch
Grade 12
Question:
A student has to match historical events viz., Dandi march, Quit India Movement and Mahatma Gandhi's assassination with the years 1948, 1930 and 1942. The student has no knowledge of the correct answer decides to match the events and years randomly. Let $E_i(0 \leq i \leq 3)$ denote the event that the student gets exactly $i$ correct answers, then which of the following is/are NOT correct?
$P(E_0) + P(E_1) = P(E_1)$
$P(E_0).P(E_1) = P(E_3)$
$P(E_0 \cap E_1) = P(E_3)$
$P(E_0) + P(E_1) + P(E_3) = 1$
Step-by-Step Solution
Key Concept: Calculate probabilities of derangements and exact matches in random permutations: P(E₀) = 2/3! = 2/6, P(E₁) = 0 (impossible to match exactly one), P(E₂) = 0 (impossible to match exactly two), P(E₃) = 1/3! = 1/6. Verify each statement algebraically using these exact probability values.
We have $P(E_0) = \frac{2^3}{3^3} = \frac{8}{27}$. For $P(E_1)$, one green marble selected in one try: $P(E_1) = \frac{\binom{3}{1} \cdot 1 \cdot 2 \cdot 2}{3^3} = \frac{3 \cdot C_1}{27}$. Similarly, $P(E_2) = \frac{\binom{3}{2} \cdot 1 \cdot 1 \cdot 2}{3^3} = \frac{3C_2}{27}$ and $P(E_3) = \frac{\binom{3}{3} \cdot 1 \cdot 1 \cdot 1}{3^3} = \frac{3C_3}{27}$.
Correct Answer: 1,2,3,4