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The Binomial Distribution

The Binomial Distribution. Section 6.2. 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. Objectives. Objective 1.

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The Binomial Distribution

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  1. The Binomial Distribution Section 6.2

  2. 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 Objectives

  3. Objective 1 Determine whether a random variable is binomial

  4. 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. What is the probability that three of you win a free hamburger? In general, if we let be the number of people out of ten that win a free hamburger. What is the probability distribution of ? In this section, we will learn that has a distribution called the binomial distribution, which is one of the most useful probability distributions. Binomial Distribution

  5. 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 represents the number of successes in 10 trials. Binomial Distribution • 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 represents the number of successes that occur. • Notation: = number of trials, = probability of a success

  6. Decide if each of the following represents a binomial experiment: • A fair coin is tossed ten times. Let be the number of times the coin lands heads. • This is a binomial experiment. Each toss of the coin is a trial. There are two possible outcomes, heads and tails. Since 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. • Five basketball players each attempt a free throw. Let be the number of free throws made. • 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. • Ten cards are in a box. Five are red and five are green. Three of the cards are drawn at random. Let be the number of red cards drawn. • This is not a binomial experiment because the trials are not independent. Example – Binomial Experiment

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

  8. Consider the binomial experiment of tossing 3 times a biased coin that has probability 0.6 of coming up heads. Let be the number of heads that come up. If we want to compute (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 (HHT) = (0.6)(0.6)(0.4) = (0.6)2(0.4). Similarly, we find that (HTH) = (THH) = (0.6)2(0.4). Now, (2) = (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 successes in trials, which is . The number 0.6 is the success probability which has an exponent of 2, the number of successes . The number 0.4 is the failure probability which has an exponent of 1, which is the number of failures, . The Binomial Probability Distribution In general, for a binomial random variable , The possible values of the random variable are 0, 1, …, .

  9. Objective 3 Compute binomial probabilities (Hand Computation)

  10. The Pew Research Center reported in June 2013 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. Example – Binomial Probabilities

  11. Solution: Note that = 15 and = 0.3. Find the probability that exactly four of the sampled people own a tablet computer. We use the binomial probability distribution with = 4: Example – Binomial Probabilities

  12. 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)15-0 + 15C1∙(0.3)1 ∙(1 – 0.3)15-1+ 15C2∙(0.3)2∙(1 – 0.3)15-2 = 0.0047+0.0305 + 0.0916 = 0.127 Example – Binomial Probabilities

  13. 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)15-0 +15C1∙(0.3)1∙(1 – 0.3)15-1 = 0.0047+0.0305 = 0.035 Now, use the Rule of Complements: Example – Binomial Probabilities

  14. 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)15-1 + 15C2∙(0.3)2∙(1–0.3)15-2 + 15C3∙(0.3)3∙(1–0.3)15-3 + 15C4∙(0.3)4∙(1–0.3)15-4 = 0.0305 + 0.0916 + 0.1700 + 0.2186 = 0.511 Example – Binomial Probabilities

  15. Objective 3 Compute binomial probabilities (Tables)

  16. A binomial table can be used to compute binomial probabilities. The Pew Research Center reported in June 2013 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 five of the sampled people own a tablet computer. Find the probability that fewer than four of the people own a tablet computer. Find the probability that the number of people who own a tablet computer is between 6 and 8, inclusive. Example – Binomial Table

  17. Solution: Note that = 15 and = 0.3. Find the probability that exactly five of the sampled people own a tablet computer. = 0.206 Example – Binomial Probabilities

  18. Find the probability that fewer than four of the people own a tablet computer. Example – Binomial Probabilities

  19. Find the probability that the number of people who own a tablet computer is between 6 and 8, inclusive. Example – Binomial Probabilities

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

  21. 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 is equalto a specific value, . The binomcdf command is used when finding the probability that the binomial random variable is less than or equal to a specified value, . Binomial Probabilities on the TI-84 PLUS

  22. binompdf To compute the probability that the random variable equals the value given the parameters and , use the binompdf command with the following format: binompdf(n,p,x) binomcdfTo compute the probability that the random variable is less than or equal to the value given the parameters and , use the binomcdf command with the following format: binomcdf(n,p,x) Binomial Probabilities on the TI-84 PLUS

  23. The Pew Research Center reported in June 2013 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. Example – Binomial Probabilities

  24. Solution: Note that = 15 and = 0.3. Find the probability that exactly four of the sampled people own a tablet computer. Since we are finding the probability that equals4, we use the binompdfcommand with = 15, = 0.3, and = 4. We find the probability that exactly four people own a tablet computer is 0.219. Example – Binomial Probabilities

  25. Find the probability that fewer than three of the people own a tablet computer. The binomcdfcommand computes the probability that there are less than or equal to 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 that fewer than three of the people own a tablet computer is 0.1268. Example – Binomial Probabilities

  26. 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 commandbinomcdf(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 0.965. Example – Binomial Probabilities

  27. Find the probability that the number of people who own a tablet computer is between 1 and 4, inclusive. Because (Between 1 and 4) = (4 or less) – (0), we can find the probability using the commands binomcdf(15, 0.30, 4) – binompdf(15, 0.30, 0). Theresult is approximately 0.511. Example – Binomial Probabilities

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

  29. Let be a binomial random variable with trials and success probability . Then the mean of is The variance of is = The standard deviation of is Mean, Variance, and Standard Deviation

  30. The probability that a new car of a certain model will require repairs during the warranty period is 0.15. A particular dealership sells 25 such cars. Let be the number that will require repairs during the warranty period. Find the mean and standard deviation of . Solution:There are = 25 trials, with success probability = 0.15. The mean is The standard deviation is Example

  31. 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 You Should Know…

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