Permutation - independent events
Task number: 3544
Let \(\pi\) be a random permutation of the set of numbers \({1, 2, …, 100}\). Let \(A_i\) be an event that occurs when \(\pi(i) = i\). Are the events \(A_1\) and \(A_2\) independent?
Solution
The event \( A_1 \) corresponds to permutations of the remaining \( 99 \) elements, ie: \(P(A_1)=P(\pi(1)=1)=\frac{99!}{100!}=\frac{1}{100}\).
By analogy we get: \( P (A_2) = \frac{1}{100} \).
The intersection of the events \( A_1 \) and \( A_2 \) corresponds to permutations of the remaining \( 98 \) elements, i.e. \(P(A_1\cap A_2)=P(\pi(1)=1 \land \pi(2)=2)=\frac{98!}{100!}=\frac{1}{9{,}900}\).
Because \(P(A_1)P(A_2)=\frac{1}{10{,}000}\ne \frac{1}{9{,}900}=P(A_1\cap A_2)\), the events \( A_1 \) and \( A_2 \) are not independent.
Answer
The events \( A_1 \) and \( A_2 \) are dependent.
