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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.
P ( X >10)
A 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.
P ( X >3)
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.
a. failure
b. If we are interested in the number of students who do their homework on time, then how do we define X ?
b.
X = the number of statistics students who do their homework on time
c. What values does x take on?
c. 0, 1, 2, …, 50
d. What is a "failure," in words?
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 ?
e.
q = 0.30
f. The words "at least" translate as what kind of inequality for the probability question P ( X ____ 40).
f. greater than or equal to (≥)
The probability question is
P (
X ≥ 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.
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.
X ~ B ( n , p )
Read this as " X is a random variable with a binomial distribution." The parameters are n and p ; n = number of trials, p = probability of a success on each trial.
P ( X = x )= $\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$
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