Question 1

When sampling is done without replacement, the binomial distribution does not give exact probabilities
because the trials are not independent.

Accepted Answer

When sampling without replacement, the probability of each trial depends on the outcomes of previous trials, making them not independent. Therefore, the binomial distribution cannot give exact probabilities in such cases.

Question 2

In a binomial experiment, the outcomes of each trial must be dependent on each other.

Accepted Answer

In a binomial experiment, the outcomes of each trial are assumed to be independent of each other.

Question 3

The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called
a binomial distribution.

Accepted Answer

This statement is true.

Question 4

$\text { Using the probability distribution listed, the mean would be } 1.6 .$
$\begin{array}{ll|l|l|l|}
\hline x & 0 & 1 & 2 & 3 \
\hline P x & 0.2 & 0.1 & 0.3 & 0.4 \
\hline
\end{array}$

Accepted Answer

The expected value or mean of a probability distribution is calculated by multiplying each possible outcome by its probability and summing them up. So, in this case, the mean would be:

$$E(X) = 0(0.2) + 1(0.1) + 2(0.3) + 3(0.4) = 1.7$$

Therefore, the statement that the mean would be 1.6 is false.

Question 5

If the standard deviation of a probability distribution is 1.71, the variance is 1.31.

Accepted Answer

The variance is equal to the square of the standard deviation, so the correct formula is variance = standard deviation^2. Using this formula, if the standard deviation is 1.71, then the variance would be (1.71)^2 = 2.92, not 1.31. Therefore, the statement is false.

Question 6

The probability of a success must remain the same for each trial in a binomial experiment.

Accepted Answer

This is one of the defining characteristics of a binomial experiment.

Question 7

The probability of observing 50 heads in 100 tosses of a fair coin is less likely than observing 500 heads
in 1000 tosses.

Accepted Answer

The probability of observing 50 heads in 100 tosses of a fair coin is higher than observing 500 heads in 1000 tosses due to the law of large numbers and the binomial distribution's variance.

Question 8

If there are 20 typographical errors randomly distributed in a 250-page document, the probability that a
given page contains exactly two errors is $0.3 \%$

Accepted Answer

The probability of a page containing exactly two errors is given by the binomial distribution:$$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$where n is the number of trials, k is the number of successes, p is the probability of success, and (1-p) is the probability of failure.In this case, n = 20, k = 2, p = 0.003, and (1-p) = 0.997. Plugging these values into the formula, we get:$$P(X = 2) = \binom{20}{2}(0.003)^2(0.997)^{18} = 0.003$$Therefore, the probability of a page containing exactly two errors is 0.3%, which is the same as the given probability.

Question 9

Discrete variables have values that can be measured.

Accepted Answer

Discrete variables have values that are distinct and separate (countable), not measured on a continuous scale.

Question 10

If the standard deviation of a probability distribution is 6.25, the variance is 2.30.

Accepted Answer

The formula for variance is the square of the standard deviation. Therefore, if the standard deviation is 6.25, the variance would be (6.25)^2 which equals 39.06. So, the statement in the question is false.

Question 11

$\text { The following distribution is not a probability distribution }$ because  
$\begin{array}{|l|c|c|c|c|c|}
\hline \mathrm{x} & -5 & -4 & -3 & -2 & -1 \
\hline \mathrm{P}(\mathrm{x}) & 0.18 & 0.26 & -0.21 & 0.46 & 0.31 \
\hline
\end{array}$
A) A values of x is negative
B) A probability is negative
C) The probability values do not add to 1
D) The probability values are not discrete

Accepted Answer

A probability is negative

Question 12

$$\text { The following distribution is not a probability distribution }$$ because 
$\begin{array}{|l|c|c|c|c|c|}
\hline \mathrm{x} & -5 & -4 & -3 & -2 & -1 \
\hline \mathrm{P}(\mathrm{x}) & 0.15 & 0.16 & 0.44 & 0.12 & 0.26 \
\hline
\end{array}$

A) The values of x are negative
B) The probability values are not increasing
C) The probability values do not add to 1
D) The probability values are not discrete

Accepted Answer

The probability values do not add up to 1.

Question 13

By computing the binomial mean and variance and using the empirical rule, one finds that the probability
of obtaining between 40 and 60 heads in 100 tosses of a fair coin is approximately 95%.

Accepted Answer

This statement is true. The binomial mean for n trials with probability of success p is given by np, and the variance is given by np(1-p). For 100 tosses of a fair coin, p=0.5, so the mean is 50 and the variance is 25. Using the empirical rule, we know that approximately 95% of the data falls within 2 standard deviations of the mean. In this case, the standard deviation is the square root of the variance, which is 5. So, we want to find the probability of obtaining between 40 and 60 heads, which is within 2 standard deviations of the mean. Using a normal approximation to the binomial distribution, we can calculate this probability to be approximately 0.9545 or 95%.

Question 14

Continuous random variables are obtained from data that can be measured rather than counted.

Accepted Answer

This statement is true. Continuous random variables represent data that can take on any value within a specified range or interval, and are typically obtained through measurement rather than counting. Examples include length, weight, temperature, and time. In contrast, discrete random variables are obtained from data that can only take on specific values (such as integers) and are typically obtained through counting or categorizing. Examples include the number of cars sold in a week, the number of people in a room, or the type of flower in a garden.

Question 15

The sum of the probabilities of all the events in the sample space must equal 1.

Accepted Answer

This is a fundamental property of probability - the probability of all possible outcomes in a sample space must add up to 1.

Question 16

$$\begin{array}{l}
\text { What probability value would be needed to complete the following probability distribution? }\
\begin{array} { | l | c | c | c | c | c | } 
\hline \mathrm { x } & - 2 & - 1 & 0 & 1 & 2 \
\hline \mathrm { P } ( \mathrm { x } ) & 0.16 & 0.17 & & 0.12 & 0.31 \
\hline
\end{array}
\end{array}$$
A) 0.12
B) 0.24
C) 0.41
D) 0.76

Accepted Answer

The sum of all probabilities in a probability distribution must equal 1. Adding the given probabilities: 0.16 + 0.17 + 0.12 + 0.31 = 0.76. Therefore, the missing probability is 1 - 0.76 = 0.24.

Question 17

The Poisson distribution is used when n is small and p is large.

Accepted Answer

The Poisson distribution is used when the number of trials or events is large and the probability of success is small. It is the opposite of the binomial distribution, which is used when n is small and p is large.

Question 18

$$\begin{array}{l}
\text { What probability value would be needed to complete the following probability distribution? }\
\begin{array} { | l | c | c | c | c | c | } 
\hline \mathrm { x } & - 2 & - 1 & 0 & 1 & 2 \
\hline \mathrm { P } ( \mathrm { x } ) & 0.16 & 0.17 & & 0.12 & 0.31 \
\hline
\end{array}
\end{array}$$
A) 0.15
B) 0.25
C) 0.10
D) There is no value that would make this a probability distribution

Accepted Answer

The answer of \[\begin{array}{l}
\text { What probability value would be...

Question 19

In a multinomial distribution, events must be mutually exclusive.

Accepted Answer

Yes, events in a multinomial distribution must be mutually exclusive, meaning that they cannot occur simultaneously. This is why the total probability of all possible outcomes must equal 1.

Question 20

In binomial experiments, the outcomes are usually classified as successes or failures.

Accepted Answer

In binomial experiments, the outcomes are indeed classified as successes or failures.

Quiz 4: Discrete Probability Distributions

When sampling is done without replacement, the binomial distribution does not give exact probabilities because the trials are not independent.

In a binomial experiment, the outcomes of each trial must be dependent on each other.

The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution.

$\text { Using the probability distribution listed, the mean would be } 1.6 .$ $\begin{array}{ll|l|l|l|}\hline x & 0 & 1 & 2 & 3 \\\hline P x & 0.2 & 0.1 & 0.3 & 0.4 \\\hline\end{array}$

If the standard deviation of a probability distribution is 1.71, the variance is 1.31.

The probability of a success must remain the same for each trial in a binomial experiment.

The probability of observing 50 heads in 100 tosses of a fair coin is less likely than observing 500 heads in 1000 tosses.

If there are 20 typographical errors randomly distributed in a 250-page document, the probability that a given page contains exactly two errors is $0.3 \%$

Discrete variables have values that can be measured.

If the standard deviation of a probability distribution is 6.25, the variance is 2.30.

$\text { The following distribution is not a probability distribution }$ because $\begin{array}{|l|c|c|c|c|c|}\hline \mathrm{x} & -5 & -4 & -3 & -2 & -1 \\\hline \mathrm{P}(\mathrm{x}) & 0.18 & 0.26 & -0.21 & 0.46 & 0.31 \\\hline\end{array}$

$\text { The following distribution is not a probability distribution }$ because $\begin{array}{|l|c|c|c|c|c|}\hline \mathrm{x} & -5 & -4 & -3 & -2 & -1 \\\hline \mathrm{P}(\mathrm{x}) & 0.15 & 0.16 & 0.44 & 0.12 & 0.26 \\\hline\end{array}$

By computing the binomial mean and variance and using the empirical rule, one finds that the probability of obtaining between 40 and 60 heads in 100 tosses of a fair coin is approximately 95%.

Continuous random variables are obtained from data that can be measured rather than counted.

The sum of the probabilities of all the events in the sample space must equal 1.

The Poisson distribution is used when n is small and p is large.

In a multinomial distribution, events must be mutually exclusive.

In binomial experiments, the outcomes are usually classified as successes or failures.

Quiz 4: Discrete Probability Distributions

When sampling is done without replacement, the binomial distribution does not give exact probabilities because the trials are not independent.

In a binomial experiment, the outcomes of each trial must be dependent on each other.

The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution.

If the standard deviation of a probability distribution is 1.71, the variance is 1.31.

The probability of a success must remain the same for each trial in a binomial experiment.

The probability of observing 50 heads in 100 tosses of a fair coin is less likely than observing 500 heads in 1000 tosses.

If there are 20 typographical errors randomly distributed in a 250-page document, the probability that a given page contains exactly two errors is 0.3%0.3 \%0.3%

Discrete variables have values that can be measured.

If the standard deviation of a probability distribution is 6.25, the variance is 2.30.

By computing the binomial mean and variance and using the empirical rule, one finds that the probability of obtaining between 40 and 60 heads in 100 tosses of a fair coin is approximately 95%.

Continuous random variables are obtained from data that can be measured rather than counted.

The sum of the probabilities of all the events in the sample space must equal 1.

The Poisson distribution is used when n is small and p is large.

In a multinomial distribution, events must be mutually exclusive.

In binomial experiments, the outcomes are usually classified as successes or failures.

If there are 20 typographical errors randomly distributed in a 250-page document, the probability that a given page contains exactly two errors is $0.3 \%$