Non-Empty letter box

Probability Level 4

Five letters are put in five identical letter boxes, the probability that no letter box is empty is

\dfrac{1}{42}

None of these

\dfrac{1}{40}

\dfrac{1}{51}

\dfrac{1}{52}

\dfrac{1}{53}

\dfrac{1}{31}

\dfrac{1}{66}

1 solution

Akhil Bansal
Jan 6, 2016

$\large \text{Probability}= \dfrac{\color{#3D99F6}{\text{Favourable Ways}}}{\color{#20A900}{\text{Total Ways}}}$ $\color{#3D99F6}{\text{Favourable Ways}}$ = No letter box is empty = $\color{#3D99F6}{1}$ way.

$\color{#20A900}{\text{Total ways}}$ = No letter box is empty + 1 letter box is empty + .... + 4 letter boxes are empty. $= \dbinom{5}{5} + \dbinom{5}{4} \times 1 + \dbinom{5}{3} \times 2 + \dbinom{5}{2} \times 2 + \dbinom{5}{1}$ $\large = 1 + 5 + 20 + 20 + 5 = \color{#20A900}{51}$ Hence, Probability = $\dfrac{\color{#3D99F6}{1}}{\color{#20A900}{51}}$

Moderator note:

The phrasing of this problem is bad, because you are making an implicit assumption on the events that are of uniform probability.

For example, even though the letter boxes are identical, I would think that the uniform probability is on posting of a particular letter into a letterbox , instead of all possible events of how the letters are placed into the identical letterbox .

As such, I disagree with your solution. Equivalently, what you are saying is that "In a lottery, there are only 2 outcomes of win or lose. Hence we win with 50%".

Non-Empty letter box

1 solution

Moderator note:

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