How many boxes will be empty?

Question

$150$ balls randomly put into $100$ boxes, each ball could be put into any of these 100 boxes with same probability, after that, on average, how many boxes will be empty? No calculator. Choose one of the following:

A 0-10
B 10-20
C 20-30
D 30-40
E 40-50
F 50-60
G 60-70
I 70-80
J 80-90
K 90-100

Answer 1

Each box gets $1.5$ balls on average and I would guess a Poisson distribution is a pretty good fit. For a Poisson distribution with expected value $\lambda$ the probability of zero is $\exp(-\lambda)$, so we need to calculate $100 \exp(-1.5)$. The square root of $e$ is between $1.6=\sqrt {2.56}$ and $1.7=\sqrt {2.89}$, so $\exp (-1.5) \approx \frac 1{1.6 \cdot 2.7} =\frac 1{4.32}$ This gives $100 \exp (-1.5) \approx 23-24$ and I will choose $20-30$

Checking with a calculator, $100 \exp (-1.5)$ is in fact $22.3$, a little lower than I guessed.

Answer 2

Perhaps you need to calculate somehow the probability of $n$ boxes to be empty $( 0 \leq n \leq 99)$ and then to calculate the expected value.