Three Properties of Binomial Distributions

In order for an experiment to be considered a binomial distribution, it must satisfy three properties:

1. There is a fixed number of trials, each with 2 outcomes.
2. The ‘trial’ outcomes are statistically independent.
3. The probability, [latex]p[/latex], of a ‘success’ is constant from trial to trial.

Two Ways of Calculating the Probability of ‘x‘ Successes

If we want to calculate the probability of exactly [latex]x[/latex] successes occurring, there are two ways:

1. Using the formula: [latex]P(x) = {}_nC_x \cdot p^{x}\cdot (1 - p)^{n-x}[/latex]
2. Using Excel: = BINOM.DIST([latex]x[/latex], [latex]n[/latex], [latex]p[/latex], 0)

two Parameters of Binomial Distributions

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

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

What are trials and successes?

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.

Exploring the properties of Binomial distributions (ExeRCISE)

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.

Calculating Probabilities Using the Binomial Formula (EXAMPLE)

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, [latex]n[/latex],[latex]p[/latex], and [latex]x[/latex]. Recall that [latex]C[/latex] stands for Combination and has no numeric value. We also know that [latex]n=10[/latex] and [latex]p=0.3[/latex] are the parameters, and we want to find the probability that [latex]x=4[/latex].

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

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.

Calculating Probabilities Using the Binomial Formula (EXErcise)

We can do the same steps as the previous example to find [latex]P(x=0)[/latex], [latex]P(x=1)[/latex], [latex]P(x=2)[/latex], … , [latex]P(x=10)[/latex]. 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 [latex]n[/latex]=10 and [latex]p[/latex]=0.3

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

You try: Solve for the probabilities below:

Key Takeaways (EXERCISE)

Your Own Notes (EXERCISE)

• Are there any notes you want to take from this section? Is there anything you’d like to copy and paste below?
• These notes are for you only (they will not be stored anywhere)
• Make sure to download them at the end to use as a reference