Answer :
The probability that at least one box contains the ball with matching label is [tex](\frac{1}{100} )^{100}[/tex].
Given that:-
Number of labeled boxes = 100
Number of labeled balls = 100
We have to find the probability that at least one box contains the ball with matching label.
We know that,
Probability that at least one box contains the ball with matching label + Probability that no box contains the ball with matching label = 1
Hence, we can write,
Probability that at least one box contains the ball with matching label = 1 - Probability that no box contains the ball with matching label.
Hence,
Probability that no box contains the ball with matching label = [tex]\frac{99^{100}}{100^{100}} = (\frac{99}{100} ) ^{100}[/tex]
Therefore,
Probability that at least one box contains the ball with matching label =
1 - [tex](\frac{99}{100} ) ^{100}[/tex] = [tex](\frac{100}{100} )^{100}-({\frac{99}{100} })^{100}=(\frac{1}{100} )^{100}[/tex].
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