There are only two parameters needed to completely ‘determine’ a binomial distribution:

• $n$  = the number of trials
• $p$ = the probability of success for each event/trial.

Trials

A ‘trial’ can be just about anything:

• The flip of a coin, in which case the 2 outcomes (heads or tails).
• A salesman calling on her clients, and making a sale or not.
• The roll of a die (where a certain number is rolled or not)
• In general, we call the 2 outcomes ‘successes’ and ‘failures’

Successes

A ‘success’ can be just about anything:

• Getting heads when flipping a coin
• Rolling a 6 when rolling dice
• Making a sale

Be Consistent

Just be sure – when talking about trials and successes related to binomial problems:

• Be sure to be consistent
• If you define success as rolling a 6,
• Be sure to use the probability of rolling as 6 as the probability of success.

In this first example we will review the 3 properties of binomial distributions. In the next example, we will calculate a probability related to the example given below.

Example 24.1.1

Problem Setup: A salesman calls on 10 clients everyday. 30% of all her calls in the past resulted in sales.

Question: Is this a Binomial experiment?

You Try: Let us find out by going through all the 3 basic characteristics.

Conclusion: Because all three properties of the binomial distribution are satisfied, this is indeed a binomial distribution.

Now that we know the situation given in the previous example is a binomial experiment, let us revisit this example when practicing using the binomial formula.

Example 24.1.2

Problem Setup: Let us suppose the salesman is still calling 10 clients per day and the probability that she will make a sale with each client is 0.3.

Question: What is the probability that 4 of her 10 calls in a day will result in sales?

Solution: If we look at this formula first, we see that there are only 3 variables, $n$,$p$, and $x$. Recall that $C$ stands for Combination and has no numeric value. We also know that $n=10$ and $p=0.3$ are the parameters, and we want to find the probability that $x=4$.

$$P(x=4) = {}_{10}C_4\cdot 0.3^4\cdot (1 – 0.3)^{10-4}={}_{10}C_4\cdot 0.3^4\cdot (0.7)^{6}$$

Let us work out each of the 3 factors individually:

$${}_{10}C_4=\frac{10!}{4!(10-4)!}=\frac{10!}{4!}{6!} =210$$

$$0.3^4 = 0.0081$$

$$0.7^6=0.0117649$$

So:

$$P(x=4)= 201\times 0.0081 \times 0.117649 = 0.200120949$$

Conclusion: There is a 20% chance that she will make exactly 4 sales in a day.

We can do the same steps as the previous example to find $P(x=0)$, $P(x=1)$, $P(x=2)$, … , $P(x=10)$. The sum of these 11 probabilities must, of course, equal 1.

Example 24.1.3

Problem Setup: We will continue with example 24.1 where $n$=10 and $p$=0.3

Question: Can you calculate the probabilities in the exercises below?

You try: Solve for the probabilities below:

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