A more valuable probability density function with many applications is the binomial distribution. This distribution will compute probabilities for any binomial process. A binomial process, often called a Bernoulli process after the first person to fully develop its properties, is any case where there are only two possible outcomes in any one trial, called successes and failures. It gets its name from the binary number system where all numbers are reduced to either 1's or 0's, which is the basis for computer technology and CD music recordings.
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable \(X\) = the number of successes obtained in the \(n\) independent trials.
The mean, \(\mu\), and variance, \(\sigma^2\), for the binomial probability distribution are \(\mu = np\) and \(\sigma^2 = npq\). The standard deviation, \(\sigma\), is then \sigma = \(\sqrt\).
Any experiment that has characteristics three and four and where \(n = 1\) is called a Bernoulli Trial (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.
Suppose 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. Here, if you define \(X\) as the number of wins, then \(X\) takes on the values 0, 1, 2, 3, . 20. The probability of a success is \(p = 0.55\). The probability of a failure is \(q = 0.45\). The number of trials is \(n = 20\). The probability question can be stated mathematically as \(P(x = 15)\)
A 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. Find the \(P(X=12)\) using the binomial PDF
A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than ten heads? Let \(X\) = the number of heads in 15 flips of the fair coin. \(X\) takes on the values 0, 1, 2, 3, . 15. Since the coin is fair, \(p = 0.5\) and \(q = 0.5\). The number of trials is \(n = 15\). State the probability question mathematically.
Answer
Approximately 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.
a. 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.
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b. If we are interested in the number of students who do their homework on time, then how do we define \(X\)?
Answer
b. \(X\) = the number of statistics students who do their homework on time
c. What values does \(x\) take on?
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d. What is a "failure," in words?
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d. Failure is defined as a student who does not complete his or her homework on time.
The probability of a success is \(p = 0.70\). The number of trials is \(n = 50\).
e. If \(p + q = 1\), then what is \(q\)?
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f. The words "at least" translate as what kind of inequality for the probability question \(P(x\) ____ 40).
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f. greater than or equal to (\(\geq\))
The probability question is \(P(x \geq 40)\).
Sixty-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
During the 2013 regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league. DeAndre scored with 61.3% of his shots. Suppose you choose a random sample of 80 shots made by DeAndre during the 2013 season. Let \(X\) = the number of shots that scored points.
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