How do you calculate $P(A)$ for the event that at least one box is empty when placing 4 distinct balls into 4 distinct boxes?

Conditions

• The balls are distinct.
• The boxes are distinct.
• Each ball must be placed into one box.
• Empty boxes are allowed in the total sample space.

Reasoning, step by step

1. Determine the total number of outcomes in the sample space: $4^4$.
2. Identify the complementary event $A^c$: no box is empty.
3. Calculate the number of outcomes for $A^c$: since 4 balls go into 4 boxes with none empty, each box has exactly 1 ball, resulting in 4! permutations.
4. Compute $P(A^c) = 4$! / $4^4$.
5. Use the complement rule: $P(A) = 1 - P(A^c)$.

Example

The video shows $P(A) = 1 - A_4^4 / 4^4$, which simplifies to $1 - 24/256 = 29/32$.

Common misconceptions

• Trying to sum the probabilities of exactly 1, 2, 3, and 4 empty boxes directly.
• Forgetting that the total sample space size is $4^4$, not 4! or another value.

Watch the explanation

Connected concepts

Explore next

• What is the value of $A_4^4$?
• How does this $P(A)$ serve as the denominator in the conditional probability formula?

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