Create an example of an application oriented binomial experiment. You are to give the scenario which should include an “n” and “p”.

Suppose we have some event that can either occur or not
occur. For example, if we flip a coin, it is heads or it isn't heads. We'll say that the
probability of the event occurring is
p.

Our experiment will be flipping a
coin. This is a fair coin, so the probability that the flip will be heads is
p=0.5.

We will proceed to flip the
coin n times. We'll say that `X_n` represents
thenth flip, and is 1 if the coin was heads, and zero
otherwise.

Then let `Y = X_1 + ... +
X_n`

That is, Y is the total number of heads
that occurred during our experiment ofn
flips.

Y then follows a Binomial Distribution
with n trials having
probability p.

In a binomial
distribution, the probability of getting exactly k heads
during ntrials is given by the
formula

`P(Y = k) =
((n),(k))p^k(1-p)^{n-k}`

In our example, suppose we wanted
to know the probability of flipping a coin 20 times and having exactly 5 heads. We
plug n=20, p=0.5, k=5 into the
above:

`P(Y = 5) = ((20),(5))0.5^5(1-0.5)^15 approx
0.015`

We evaluate the above, and find that the probability
of getting 5 heads out of 20 flips is approximately
0.015.

The important thing to take out of this example is
that a binomial distribution can be used to model a series of events that either do or
do not occur (with some probability).

Here are some more
examples:

1. The probability of rolling a 6 is 1/6. What is
the probability of rolling 3 sixes in 5 trials (n=5,
p=1/6)

2. The probability of me missing my bus to work is
1/100. What is the probability I'll miss the bus two days this work week? (n=5,
p=1/100)

3. The probability I'll get a phone call during
any given hour is 1/5. What's the probability I'll get 10 phone calls during the next 8
hours? (n=8, p=1/5).