1. Essential Statistics 2E
William Navidi and Barry Monk

2. The Binomial Distribution
Section 5.2

3. Objectives Determine whether a random variable is binomial
Determine the probability distribution of a binomial random variable
Compute binomial probabilities
Compute the mean and variance of a binomial random variable

4. Objective 1
Determine whether a random variable is binomial

5. Binomial Distribution
Suppose that your favorite fast food chain is giving away a coupon with every purchase of a meal. Twenty percent of the coupons entitle you to a free hamburger, and the rest of them say “better luck next time.” Ten of you order lunch at this restaurant. Suppose we want to know the probability that three of you win a free hamburger? In general, if we let 𝑋 be the number of people out of ten that win a free hamburger. What is the probability distribution of 𝑋? In this section, we will learn that 𝑋 has a distribution called the binomial distribution, which is one of the most useful probability distributions.

6. Conditions for a Binomial Distribution
In the problem just described, each time we examine a coupon, we call it a “trial,” so there are 10 trials. When a coupon is good for a free hamburger, we will call it a “success.” The random variable 𝑋 represents the number of successes in 10 trials.
A random variable that represents the number of successes in a series of trials has a probability distribution called the binomial distribution. The conditions are:
A fixed number of trials are conducted.
There are two possible outcomes for each trial. One is labeled “success” and the other is labeled “failure.”
The probability of success is the same on each trial.
The trials are independent. This means that the outcome of one trial does not affect the outcomes of the other trials.
The random variable 𝑋 represents the number of successes that occur.
Notation: 𝑛 = number of trials, 𝑝 = probability of a success

7. Example 1: Binomial Experiment
A fair coin is tossed ten times. Let 𝑋 be the number of times the coin lands heads. Is this a binomial experiment? This is a binomial experiment. Each toss of the coin is a trial. There are two possible outcomes, heads and tails. Since 𝑋 represents the number of heads, heads counts as a success. The trials are independent, because the outcome of one coin toss does not affect the other tosses.

8. Example 2: Binomial Experiment
Five basketball players each attempt a free throw. Let 𝑋 be the number of free throws made. Is this a binomial experiment? This is not a binomial experiment. The probability of success (making a shot) differs from player to player, because they will not all be equally skilled at making free throws.

9. Example 3: Binomial Experiment
Ten cards are in a box. Five are red and five are green. Three of the cards are drawn at random. Let 𝑋 be the number of red cards drawn. Is this a binomial experiment? This is not a binomial experiment because the trials are not independent.

10. Objective 2
Determine the probability distribution of a binomial random variable

11. The Binomial Probability Distribution
Consider the binomial experiment of tossing 3 times a biased coin that has probability 0.6 of coming up heads. Let 𝑋 be the number of heads that come up. If we want to compute 𝑃(2), the probability that exactly 2 of the tosses are heads, there are 3 arrangements of two heads in three tosses: HHT, HTH, THH. The probability of HHT is 𝑃(HHT) = (0.6)(0.6)(0.4) = (0.6)2(0.4). Similarly, we find that 𝑃(HTH) = 𝑃(THH) = (0.6)2(0.4). Now, 𝑃(2) = 𝑃(HHT or HTH or THH) = 3(0.6)2(0.4), by the Addition Rule. Examining this result, we see the number 3 represents the number of arrangements of two successes (heads) and one failure (tails). In general, this number will be the number of arrangements of 𝑥 successes in 𝑛 trials, which is 𝑛𝐶𝑥. The number 0.6 is the success probability 𝑝 which has an exponent of 2, the number of successes 𝑥. The number 0.4 is the failure probability 1−𝑝 which has an exponent of 1, which is the number of failures, 𝑛−𝑥.

12. The Binomial Probability Distribution Formula
In general, for a binomial random variable 𝑋, 𝑷 𝒙 =𝒏𝑪𝒙∙ 𝒑 𝒙 ∙ 𝟏−𝒑 𝒏−𝒙 The possible values of the random variable 𝑋 are 0, 1, …, 𝑛.

13. Objective 3*
Compute binomial probabilities
*(Hand Computation)

14. Example: Binomial (Hand Computation)
The Pew Research Center reported in a recent year that approximately 30% of U.S. adults own a tablet computer such as an iPad, Samsung Galaxy Tab, or Kindle Fire. Suppose a simple random sample of 15 people is taken. Use the binomial probability distribution to find the following probabilities.
Find the probability that exactly four of the sampled people own a tablet computer.
Find the probability that fewer than three of the people own a tablet computer.
Find the probability that more than one person owns a tablet computer.
Find the probability that the number of people who own a tablet computer is between 1 and 4, inclusive.

15. Example: Part a) (Hand Comp.)
Note that 𝑛 = 15 and 𝑝 = 0.3.
Find the probability that exactly four of the sampled people own a tablet computer.
We use the binomial probability distribution with 𝑥 = 4:
𝑃 4 &=15𝐶4∙ ∙ 1− −4 &= 15! 4! 15−4 ! ∙ ∙ &=1365∙ ∙ &=0.219

16. Example: Part b) (Hand Comp.)
Find the probability that fewer than three of the people own a tablet computer.
The possible numbers of people that are fewer than three are 0, 1, and 2:
P (0 or 1 or 2) = 15C0∙(0.3)0 ∙(1 – 0.3) C1∙(0.3)1 ∙(1 – 0.3) C2∙(0.3)2 ∙(1 – 0.3)15-2 =
= 0.127

17. Example: Part c) (Hand Comp.)
Find the probability that more than one person owns a tablet computer.
We use the Rule of Complements. The complement of “more than 1” is “1 or fewer” or equivalently, 0 or 1.
The probability of the 0 or 1 is:
P (0 or 1) = 15C0∙(0.3)0 ∙(1 – 0.3) C1∙(0.3)1 ∙(1 – 0.3)15-1 =
= 0.035
Now, use the Rule of Complements:
𝑃 More than 1 &=1−𝑃 0 or 1 &=1−0.035 &=0.965

18. Example: Part d) (Hand Comp.)
Find the probability that the number of people who own a tablet computer is between 1 and 4, inclusive.
Between 1 and 4 inclusive means, 1, 2, 3, or 4.
P (1 or 2 or 3 or 4) = 15C1∙(0.3)1 ∙(1–0.3) C2∙(0.3)2 ∙(1–0.3)15-2
+ 15C3∙(0.3)3 ∙(1–0.3) C4∙(0.3)4 ∙(1–0.3)15-4 =
= 0.511

19. Objective 3**
Compute binomial probabilities
**(Tables)

20. Example: Binomial (Tables)
The Pew Research Center reported in a recent year that approximately 30% of U.S. adults own a tablet computer such as an iPad, Samsung Galaxy Tab, or Kindle Fire. Suppose a simple random sample of 15 people is taken. Use the binomial probability distribution to find the following probabilities.
Find the probability that exactly four of the sampled people own a tablet computer.
Find the probability that fewer than three of the people own a tablet computer.
Find the probability that more than one person owns a tablet computer.
Find the probability that the number of people who own a tablet computer is between 1 and 4, inclusive.

21. Example: Part a) (Tables)
Note that 𝑛 = 15 and 𝑝 = 0.3.
a) Find the probability that exactly four of the sampled people own a tablet computer.
𝑃(4) = 0.219

22. Example: Part b) (Tables)
b) Find the probability that fewer than three of the people own a tablet computer.
𝑃 Fewer than 3 = 𝑃 0 +𝑃 1 +𝑃 2 = =0.128

23. Example: Part c) (Tables)
c) Find the probability that more than one person owns a tablet computer.
We use the Rule of Complements. The complement of “more than 1” is “1 or fewer” or equivalently, 0 or 1. The probability of the 0 or 1 is: P (0 or 1) = = Now, use the Rule of Complements: 𝑃 More than 1 &=1−𝑃 0 or 1 &=1−0.036 &=0.964

24. Example: Part d) (Tables)
d) Find the probability that the number of people who own a tablet computer is between 1 and 4, inclusive.
𝑃 Between 1 and 4, inclusive = 𝑃 1 +𝑃 2 +𝑃 3 +𝑃 4 = =0.511

25. Objective 3***
Compute binomial probabilities
***(TI-84 PLUS)

26. Binomial Probabilities on the TI-84 PLUS
In the TI-84 PLUS Calculator, there are two primary commands for computing binomial probabilities. These are binompdf and binomcdf. These commands are on the DISTR (distributions) menu accessed by pressing 2nd, VARS. The binompdf command is used when finding the probability that the binomial random variable 𝑋 is equal to a specific value, 𝑥. The binomcdf command is used when finding the probability that the binomial random variable 𝑋 is less than or equal to a specified value, 𝑥.

27. binompdf and binomcdf Commands
To compute the probability that the random variable 𝑋 equals the value 𝑥 given the parameters 𝑛 and 𝑝, use the binompdf command with the following format:
binompdf(n,p,x)
binomcdf To compute the probability that the random variable 𝑋 is less than or equal to the value 𝑥 given the parameters 𝑛 and 𝑝, use the binomcdf command with the following format:
binomcdf(n,p,x)

28. Example: Binomial (TI-84 PLUS)
The Pew Research Center reported in a recent year that approximately 30% of U.S. adults own a tablet computer such as an iPad, Samsung Galaxy Tab, or Kindle Fire. Suppose a simple random sample of 15 people is taken. Use the binomial probability distribution to find the following probabilities.
Find the probability that exactly four of the sampled people own a tablet computer.
Find the probability that fewer than three of the people own a tablet computer.
Find the probability that more than one person owns a tablet computer.
Find the probability that the number of people who own a tablet computer is between 1 and 4, inclusive.

29. Example: Part a) (TI-84 PLUS)
Note that 𝑛 = 15 and 𝑝 = 0.3.
Find the probability that exactly four of the sampled people own a tablet computer.
Since we are finding the probability that 𝑥 equals 4, we use the binompdf command with 𝑛 = 15, 𝑝 = 0.3, and 𝑥 = 4. We find the probability that exactly four people own a tablet computer is

30. Example: Part b) (TI-84 PLUS)
Find the probability that fewer than three of the people own a tablet computer.
The binomcdf command computes the probability that there are less than or equal to 𝑥 successes. The event “fewer than three” is equivalent to “less than or equal to two”.
We run the command binomcdf(15, 0.3, 2) to find that the probability that fewer than three of the people own a tablet computer is

31. Example: Part c) (TI-84 PLUS)
Find the probability that more than one person owns a tablet computer.
We use the Rule of Complements. The complement of “more than 1” is “1 or fewer”. We use the command binomcdf(15, 0.3, 1) to first find the probability that 1 or fewer people own a tablet computer, and then subtract this value from 1 to find the probability that more than one person owns a tablet computer. The result is approximately

32. Example: Part d) (TI-84 PLUS)
Find the probability that the number of people who own a tablet computer is between 1 and 4, inclusive.
Because 𝑃(Between 1 and 4) = 𝑃(4 or less) – 𝑃(0), we can find the probability using the commands
binomcdf(15, 0.30, 4) – binompdf(15, 0.30, 0)
The result is approximately

33. Objective 4
Compute the mean and variance of a binomial random variable

34. Mean, Variance, and Standard Deviation
Let 𝑋 be a binomial random variable with 𝑛 trials and success probability 𝑝. Then the mean of 𝑋 is 𝝁 𝒙 =𝒏𝒑 The variance of 𝑋 is 𝝈 𝒙 𝟐 =𝒏𝒑(𝟏−𝒑) The standard deviation of 𝑋 is 𝝈 𝒙 = 𝒏𝒑(𝟏−𝒑)

35. Example: Mean and Standard Deviation
The probability that a new car of a certain model will require repairs during the warranty period is A particular dealership sells 25 such cars. Let 𝑋 be the number that will require repairs during the warranty period. Find the mean and standard deviation of 𝑋. Solution: There are 𝑛 = 25 trials, with success probability 𝑝 = The mean is 𝜇 𝑥 &=𝑛𝑝= &=3.75 The standard deviation is 𝜎 𝑥 &= 𝑛𝑝 1−𝑝 = 25∙0.15 1−0.15 &=1.785

36. You Should Know . . .
How to determine whether a random variable is binomial
The notation for a binomial experiment
How to determine the probability distribution of a binomial random variable
How to compute binomial probabilities
How to compute the mean and variance of a binomial random variable