4.2 Binomial Distributions

So far we have looked at the probability of a particular outcome in an “experiment”. If I roll a die what is the outcome. Now we will consider repeated trials of the same experiment and rather then keep track of the individual outcomes, we will measure the number of times a particular outcome occurs.

Defn: Consider an experiment that consists of n, repeated, independent trials, whose outcome can be determined to be either success or failure. Let the random variable X represent the number of success out of the n trials. Then X follows a binomial distribution.

In such a distribution we denote “p” to be the probability of success, and q the probability of failure.

Important: The key things to look for to determine whether a random variable follows a binomial distribution are:

• x out of n trials

• Independent trials

• Outcome is can be measured in success or failure

Look for these signs when solving homework or test problems.

Example: Consider rolling a 6-sided die four times. We would like to know, “What is the probability of rolling exactly three 2’s?”

Let X represent the number of times a 2 is rolled. Then we can define success to be the roll of a 2, and failure to be any other roll. Then p= 1/6 and q = 5/6.

So, how can we determine the probability of rolling exactly three 2’s? First off, note that it can happen in several different ways. The successful rolls could happen all at once, spread out, all at the beginning, all at the end, etc.

We know that there are 4 rolls, 3 of which are successes (rolls of 2), each of which is indistinguishable, so there are ₄C₃ different ways to rolls exactly four 3s in ten rolls.

Note that in any set of rolls that contains exactly three 2’s, regardless of the order of the rolls there will be 3 successes and 4-3 = 1 failures. Recall p=1/6 and q=5/6, so the probability of such a set of rolls is:

.

Since there are ₄C₃ different sets of rolls with exactly four 3’s then the probability of getting exactly four 3s is:

Probability in a Binomial Distribution

Let’s now generalize this process. Let the random variable X represent the number of successes out of n independent trials. Let p be the probability of success and q the probability of failure. Then the probability of x successes out of n trials is:

Expected Value for a Binomial Distribution

The expected value of a random variable X that follows a binomial distribution is given by:

Example: The Choco-Latie Candies company makes candy-coated chocolates, 40% of which are red. The production line mixes the candies randomly and packages ten per box.

a) What is the probability that at least three candies in a given box are red?

b) What is the expected number of read candies in a box?

Solution: Let X represent the number of red candies in a box.

There are ten candies is a box so n = 10

A success will be a candy being red, so the probability of success is p = 40%.

The probability of failure is q = 60%.

1. P(X ≥3) = 1 – P(X < 3) (Indirect Method)

= 1 – P(X=0) – P(X=1) – P(X=2)

= 1 – ₁₀C₀(0.4)⁰ (0.6)¹⁰ - ₁₀C₁(0.4)¹ (0.6)⁹ - ₁₀C₂(0.4)² (0.6)²

= 0.8327

Therefore the probability of getting at least 3 red candies in a box in roughly 83%

Note: Since we see the term at least, we know to use the indirect method.

2. E[X] = n p

= (10) (0.4)

= 4

Therefore the expected number of red candies in a box is 4.

Homework: pg 384 #1-3, 5, 7, 11-13