Suppose balls are tossed 
bins until
some bin contains two balls. Besides tossing the ball, the
following points must also be considered:
• Each toss is independent.
• Each ball is likely to end up in any one of the given bin.
Consider that two balls 
and
are
tossed. Then, suppose;
Now, consider the tosses are performed 
times. then
in this case,
The total numbers of bins are. So the
expected value 
will be
.
Since, there are two balls, 
and
, and total
number of bins exists are, so the
chance of landing the balls in the same bin is.
Use this expectation in the equation (1).
Now, set the value of 
to 1 in the
equation (2).
The formula to find the roots of a quadratic equation of the
form 
is given
below:
Solve the quadratic equation (3) using the above formula to find the values of n.
Neglecting
, as the
value will be less than 1 for any values of b (the number of
bins are never be less than 1).
Hence, the expected number of balls tosses into
bins
until some bin contains two balls is, 
,
where,
denotes the number of bins.
Suppose 
denotes an
event in which all the tossed balls fall in bins, other than the
bin.
Here, balls are
tossed into bins and
every toss is not dependent to each other and also each ball is
equally likely to end up in any bin.
Calculate the probability for the event .
The expected number’s linear property is given below:
,
wheredenotes the
indicator random variable.
Now, calculate as
follows:
Hence, the expected number of empty bins is 
.
Suppose 
denotes a
random variable of bins with exactly one ball.
Here, balls are
tossed into bins and
every toss is not dependent to each other and also each ball is
equally likely to end up in any bin.
Here, denotes the
indicator random variable, then,
So,
Calculate the as
follows:
Hence, the expected number of bins with exactly one ball
is 
.