Practice problems for these concepts can be found at:

- Discrete Random Variables Solved Problems for Beginning Statistics
- Discrete Random Variables Supplementary Problems for Beginning Statistics

### Binomial Randon Variable

A *binomial random variable* is a discrete random variable that is defined when the conditions of a *binomial experiment* are satisfied. The conditions of a binomial experiment are given in Table 5.7.

The two outcomes possible on each trial are called *success* and *failure*. The probability associated with success is represented by the letter p and the probability associated with failure is represented by the letter q, and since one or the other of success or failure must occur on each trial, p + q must equal one, i.e., p + q = 1. When the conditions of the binomial experiment are satisfied, the binomial random variable X is defined to equal the number of successes to occur in the n trials. The random variable X may assume any one of the whole numbers from zero to n.

**EXAMPLE 5.11** A balanced coin is tossed 10 times, and the number of times a head occurs is represented by X. The conditions of a binomial experiment are satisfied. There are n = 10 identical trials. Each trial has two possible outcomes, head or tail. Since we are interested in the occurrence of a head on a trial, we equate the occurrence of a head with success, and the occurrence of a tail with failure. We see that p = .5 and q = .5. Also, it is clear that the trials are independent since the occurrence of a head on a given toss is independent of what occurred on previous tosses. The number of heads to occur in the 10 tosses, X, can equal any whole number between 0 and 10. X is a binomial random variable with n = 10 and p = .5.

**EXAMPLE 5.12** A balanced die is tossed five times, and the number of times that the face with six spots on it faces up is counted. The conditions of a binomial experiment are satisfied. There are five identical trials. Each trial has two possible outcomes since the face 6 turns up or a face other than 6 turns up. Since we are interested in the face 6, we equate the face 6 with success and any other face with failure. We see that p = and q = . Also, the outcomes from toss to toss are independent of one another. The number of times the face 6 turns up, X, can equal 0, 1, 2, 3, 4, or 5. X is a binomial random variable with n = 5 and p = .167

**EXAMPLE 5.13** A manufacturer uses an injection mold process to produce disposable razors. One-half of one percent of the razors are defective. That is, on average, 500 out of every 100,000 razors are defective. A quality control technician chooses a daily sample of 100 randomly selected razors and records the number of defectives found in the sample in order to monitor the process. The conditions of a binomial experiment are satisfied. There are 100 identical trials. Each trial has two possible outcomes since the razor is either defective or non-defective. Since we are recording the number of defectives, we equate the occurrence of a defective with success and the occurrence of a nondefective with failure. We see that p = .005 and q = .995. The number of defectives in the 100, X, can equal any whole number between 0 and 100. X is a binomial random variable with n = 100 and p = .005.

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