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)
Try it
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).
Try it
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.
Notation for the binomial:
B = binomial probability distribution function
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.
Binomial formula
P (
X =
x )=
$\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$
$p^{x}$ is the probability of
x successes in
x independent and identical trials. For example, if the probability of success = 0.4, the probability of five successes in five independent and identical trials is
$q^{\mathrm{n-x}}$ is the probability of
n - x failures in
n - x identical and independent trials. For example, if the probability of success = 0.4, then the probability of failure is 1 -
p = 0.6. If there are eight trials (
n = 8) with five successes (
x = 5 ), then there were three failures in the eight trials (
n - x = 8 - 5 = 3). The probability of three failures in three independent and identical trials is
$\left(\begin{array}{c}n\\ x\end{array}\right)$ represents the number of combinations of
x successes in
n trials. If there are eight trials (
n = 8) and five successes (
x = 5), then there are 56 possible ways to arrange five successes among eight trials.
The formula
P (
X =
x )=
$\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$ is the probability of
x successes in
n independent and identical trials. If there are eight independent and identical trials, the probability of five successes where
p = 0.4 is
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Source:
OpenStax, Introduction to statistics i - stat 213 - university of calgary - ver2015revb. OpenStax CNX. Oct 21, 2015 Download for free at http://legacy.cnx.org/content/col11874/1.3
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