{"id":244,"date":"2021-07-14T15:58:57","date_gmt":"2021-07-14T15:58:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/binomial-distribution\/"},"modified":"2025-01-14T15:59:08","modified_gmt":"2025-01-14T15:59:08","slug":"binomial-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/binomial-distribution\/","title":{"raw":"What are Binomial Experiments?","rendered":"What are Binomial Experiments?"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
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  • Describe the three characteristics of a binomial experiment<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nThere are three characteristics of a binomial experiment<\/strong>.\r\n
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    1. There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter [latex]n[\/latex] denotes the number of trials.<\/li>\r\n \t
    2. There are only two possible outcomes, called \"success\" and \"failure,\" for each trial. The letter\u00a0[latex]p[\/latex]\u00a0<\/em>denotes the probability of a success on one trial, and [latex]q[\/latex] denotes the probability of a failure on one trial. [latex]p+q=1[\/latex].<\/li>\r\n \t
    3. The [latex]n[\/latex] trials are independent and are repeated using identical conditions. Because the [latex]n[\/latex] trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability,\u00a0[latex]p[\/latex]\u00a0of a success and probability, [latex]q[\/latex], of a failure remain the same. For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with probability [latex]p=0.6[\/latex]. Then, [latex]q=0.4[\/latex]. This means that for every true-false statistics question Joe answers, his probability of success [latex](p=0.6)[\/latex] and his probability of failure [latex](q=0.4)[\/latex]\u00a0remain the same.<\/li>\r\n<\/ol>\r\nThe outcomes of a binomial experiment fit a binomial probability distribution<\/strong>. The random variable [latex]X=[\/latex]\u00a0the number of successes obtained in the [latex]n[\/latex] independent trials.\r\n\r\nThe mean, [latex]\\mu[\/latex], and variance, [latex]\\sigma^{2}[\/latex], for\u00a0the binomial probability distribution are [latex]\\mu=np[\/latex]\u00a0and [latex]\\sigma^{2}=npq[\/latex]. The standard deviation, [latex]\\sigma[\/latex], is then [latex]\\sigma=\\sqrt{{{n}{p}{q}}}[\/latex].\r\n\r\nAny experiment that has characteristics two and three and where [latex]n=1[\/latex] is called a Bernoulli Trial<\/strong> (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.\r\n
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      Example<\/h3>\r\nAt ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A \"success\" could be defined as an individual who withdrew. The random variable [latex]X=[\/latex] the number of students who withdraw from the randomly selected elementary physics class.\r\n\r\n<\/div>\r\n
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      Try It<\/h3>\r\nThe state health board is concerned about the amount of fruit available in school lunches. Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What would a \"success\" be in this case?\r\n\r\n[reveal-answer q=\"539268\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"539268\"]A success would be a school that offers fruit in their lunch every day.[\/hidden-answer]\r\n\r\n<\/div>\r\n
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      Example<\/h3>\r\nSuppose you play a game that you can only either win or lose. The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times.\r\n\r\n[reveal-answer q=\"328186\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"328186\"]Here, if you define [latex]X[\/latex] as the number of wins, then [latex]X[\/latex] takes on the values 0, 1, 2, 3, ..., 20. The probability of a success is [latex]p=0.55[\/latex]. The probability of a failure is [latex]q=0.45[\/latex]. The number of trials is [latex]n=20[\/latex]. The probability question can be stated mathematically as [latex]P(x=15)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
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      Try It<\/h3>\r\nA trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. State the probability question mathematically.\r\n\r\n[reveal-answer q=\"564040\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"564040\"][latex]P(x=12)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n
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      Example<\/h3>\r\nA fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than ten heads? Let [latex]X=[\/latex] the number of heads in 15 flips of the fair coin. [latex]X[\/latex] takes on the values 0, 1, 2, 3, ..., 15. Since the coin is fair, [latex]p=0.5[\/latex] and [latex]q=0.5[\/latex]. The number of trials is [latex]n=15[\/latex]. State the probability question mathematically.\r\n\r\n[reveal-answer q=\"995613\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"995613\"][latex]P(x>10)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n
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      Try It<\/h3>\r\nA fair, six-sided die is rolled ten times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically.\r\n\r\n[reveal-answer q=\"244297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"244297\"][latex]P(x>3)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n
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      Example<\/h3>\r\nApproximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.\r\n
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      1. This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.<\/li>\r\n \t
      2. If we are interested in the number of students who do their homework on time, then how do we define [latex]X[\/latex]?<\/li>\r\n \t
      3. What values does [latex]x[\/latex] take on?<\/li>\r\n \t
      4. What is a \"failure,\" in words?<\/li>\r\n \t
      5. If [latex]p+q=1[\/latex], then what is [latex]q[\/latex]?<\/li>\r\n \t
      6. The words \"at least\" translate as what kind of inequality for the probability question\u00a0[latex]P(x\\geq40)[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"872394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"872394\"]\r\n
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        1. failure<\/li>\r\n \t
        2. [latex]X=[\/latex] the number of statistics students who do their homework on time<\/li>\r\n \t
        3. 0, 1, 2, \u2026, 50<\/li>\r\n \t
        4. Failure is defined as a student who does not complete his or her homework on time. The probability of a success is [latex]p=0.70[\/latex]. The number of trials is [latex]n=50[\/latex].<\/li>\r\n \t
        5. [latex]q=0.30[\/latex]<\/li>\r\n \t
        6. greater than or equal to (\u2265)The probability question is [latex]P(x\\geq40)[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
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          Try It<\/h3>\r\nSixty-five percent of people pass the state driver's exam on the first try. A group of 50 individuals who have taken the driver's exam is randomly selected. Give two reasons why this is a binomial problem.\r\n\r\n[reveal-answer q=\"720742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"720742\"]This is a binomial problem because there is only a success or a failure, and there are a definite number of trials. The probability of a success stays the same for each trial.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n ","rendered":"
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          Learning Outcomes<\/h3>\n
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