There are n different object 1, 2, 3, ... n distributed at random in n boxes A1, A2, A3,... AN. Find the probability that two objects are placed in the boxes corresponding to their number.

Engineering

Mathematics

Conditional Probability

Question

There are n different object 1, 2, 3, ... n distributed at random in n boxes A₁, A₂, A₃,... A_N. Find the probability that two objects are placed in the boxes corresponding to their number.

$\frac{1}{2}$

$\frac{1}{3}$

$\frac{2}{3}$

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Let E_i be the event that ith object is placed in ithe box,

P (E_{i}) = 1 \cdot \frac{(n - 1)}{n!} = \frac{1}{n}, i = 1, 2 \dots n .

Now the probability that two boxes contain right objects.

P (E_{i} \cap E_{j}) = 1 . \frac{(n - 2)!}{n!}

and these two objects can be out of n objects in ⁿC₂ ways.

So the probability that two objects are in right boxes.

=^{n} C_{2} \frac{(n - 2)!}{n!} = \frac{n!}{2! (n - 2)!} \cdot \frac{(n - 2)}{n!} = \frac{1}{2} .

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