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

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### 1 Answer

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 the*nth* 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 of*n* 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 *n*trials 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).