2025 MATH 1680 FINAL EXAMS QUESTIONS WITH
ANSWERS GRADED A+ 2025/2026
The binomial probability distribution is symmetric and approximately bell shaped if - p =
0.5
The binomial probability distribution is skewed left if: - p > 0.5
Under what conditions will a binomial probability distribution be approximately bell-
shaped? - For a fixed p, as the number of trials n in a binomial experiment increases,
the probability distribution of the random variable X becomes bell-shaped.
Explain how to determine if an observation in a binomial experiment is unusual. - As a
rule of thumb, if np(1−p)≥10, the probability distribution will be approximately bell-
shaped.
When can the Empirical Rule be used to identify unusual results in a binomial
experiment? Why can the Empirical Rule be used to identify results in a binomial
experiment? - When the binomial distribution is approximately bell shaped, about 95%
of the outcomes will be in the interval from
μ−2σ and μ+2σ.
The Empirical Rule can be used to identify results in binomial experiments when
np(1−p)≥10
In sampling without replacement, the assumption of independence required for a
binomial experiment is violated. Under what circumstances can we sample without
replacement and still use the binomial probability formula to approximate probabilities? -
As a rule of thumb, if the sample size is
less than 5%
of the population size, the trials can be considered nearly independent.
Define simulation. - A technique used to recreate a random event. (can be tactile or
virtual) The goal is to measure how often a certain outcome observed.
Define a random process. - A random process represents scenarios where the outcome
of any particular trial of an experiment is unknown, but the proportion or relative
frequency a particular outcome is observed approaches a specific value
The word _________ suggests an unpredictable result or outcome. - random
Define probability. - Probability is the measure of the likelihood of a random
phenomenon or chance behavior occurring. It deals with experiments that yield random
short-term results or outcomes yet reveal long-term predictability. The long-term
proportion in which a certain outcome is observed is the probability of that outcome.
,State the Law of Large Numbers. - As the number of repetitions of a probability
experiment increases, the proportion with which a certain outcome is observed gets
closer to the probability of the outcome.
Explain the meaning of the sentence, "In a random process, the trials are memoryless."
- Trials do not recall what has happened in the past and used them to make changes on
what's going to happen in the future
In probability, what is an experiment? - In probability, an experiment is any process with
uncertain results that can be repeated. The result of any single trial of the experiment is
not known ahead of time. However, the results of the experiment over many trials
produce regular patterns that allow accurate predictions.
A(n) _______ is any collection of outcomes from a probability experiment. - event
What is a probability model? - A probability model lists the possible outcomes of a
probability experiment and each outcome's probability. A probability model must satisfy
Rules 1 and 2 of the rules of probabilities.
What is an unusual event? What cutoff points do statisticians typically use for identifying
unusual events? - An unusual event is an event that has a low probability of occurring.
Typically, an event with a probability less than 0.05 (or 5%) is considered unusual, but
this cutoff point is not set in stone. Statisticians typically use cutoff points of 0.01,0.05,
and 0.10
List the three methods in this section for determining the probability of an event. - 1. the
Empirical Method
2. the Classical Method
3. the Subjective Method
Surveys are probability experiments. Why? - Each time a survey is conducted, a
different random sample of individuals is selected. Therefore, the results of a survey are
likely to be different each time the survey is conducted because different people are
included.
The _________ method gives an approximate probability of an event by conducting a
probability experiment. The ____________ method of computing probabilities does not
require that a probability experiment actually be performed, rather it relies on counting
techniques. - empirical; classical
What requirement must be met in order to compute probabilities using the classical
method? - The classical method requires equally likely outcomes. An experiment has
equally likely outcomes when each outcome has the same probability of occurring
As the number of trials of an experiment increase, how does the empirical probability of
an event occurring compare to the classical probability of that event occurring? - The
, empirical probability will get closer to the classical probability as the number of trials of
the experiment increases due to the Law of Large Numbers. If the two probabilities do
not get closer, we may suspect that the dice are not fair.
In ______________, each individual has the same chance of being selected. Therefore,
we can use the classical method to compute the probability of obtaining a specific
sample - simple random sampling
What is a subjective probability? Explain why subjective probabilities are used. - A
subjective probability is a probability that is determined based on personal judgement.
Subjective probabilities are legitimate and are often the only method of assigning
likelihood to an outcome. For instance, a financial reporter may ask an economist about
the likelihood of the economy falling into recession next year. Again, we cannot conduct
an experiment n times to find a relative frequency. The economist must use knowledge
of the current conditions of the economy and make an educated guess about the
likelihood of recession.
Explain the Law of Large Numbers. How does this apply to gambling casinos? - As the
number of repetitions of a probability experiment increases, the proportion with which a
certain outcome is observed gets closer to the probability of the outcome. This applies
to casinos because they are able to make a profit in the long run because they have a
small statistical advantage in each game.
What does it mean for two events to be disjoint? - Two events are disjoint if they have
no outcomes in common. Another name for disjoint events is mutually exclusive events.
If two events are mutually exclusive, it means that they cannot occur at the same time.
In a Venn diagram, what does the rectangle represent? What does a circle represent? -
The rectangle represents the sample space and each circle represents an event
How can you tell from a Venn diagram that two events are not disjoint? - The circles are
drawn to overlap.
State the Addition Rule for Disjoint Events. - If E and F are disjoint (or mutually
exclusive) events, then P(E or F) = P(E) + P(F)
For disjoint events E and F, how is P(E or F) related to P(E) and P(F)? - You add the
two probabilities together so P(E or F) = P(E) + P(F)
State the General Addition Rule. - For any two events, E and R, P(E or F) = P(E) + P(F)
- P(E and F)
Explain why we subtract P(E and F) when using the General Addition Rule. - We
subtract P(E and F) in order to avoid double counting. We are subtracting the probability
corresponding to the overlapping region E and F
ANSWERS GRADED A+ 2025/2026
The binomial probability distribution is symmetric and approximately bell shaped if - p =
0.5
The binomial probability distribution is skewed left if: - p > 0.5
Under what conditions will a binomial probability distribution be approximately bell-
shaped? - For a fixed p, as the number of trials n in a binomial experiment increases,
the probability distribution of the random variable X becomes bell-shaped.
Explain how to determine if an observation in a binomial experiment is unusual. - As a
rule of thumb, if np(1−p)≥10, the probability distribution will be approximately bell-
shaped.
When can the Empirical Rule be used to identify unusual results in a binomial
experiment? Why can the Empirical Rule be used to identify results in a binomial
experiment? - When the binomial distribution is approximately bell shaped, about 95%
of the outcomes will be in the interval from
μ−2σ and μ+2σ.
The Empirical Rule can be used to identify results in binomial experiments when
np(1−p)≥10
In sampling without replacement, the assumption of independence required for a
binomial experiment is violated. Under what circumstances can we sample without
replacement and still use the binomial probability formula to approximate probabilities? -
As a rule of thumb, if the sample size is
less than 5%
of the population size, the trials can be considered nearly independent.
Define simulation. - A technique used to recreate a random event. (can be tactile or
virtual) The goal is to measure how often a certain outcome observed.
Define a random process. - A random process represents scenarios where the outcome
of any particular trial of an experiment is unknown, but the proportion or relative
frequency a particular outcome is observed approaches a specific value
The word _________ suggests an unpredictable result or outcome. - random
Define probability. - Probability is the measure of the likelihood of a random
phenomenon or chance behavior occurring. It deals with experiments that yield random
short-term results or outcomes yet reveal long-term predictability. The long-term
proportion in which a certain outcome is observed is the probability of that outcome.
,State the Law of Large Numbers. - As the number of repetitions of a probability
experiment increases, the proportion with which a certain outcome is observed gets
closer to the probability of the outcome.
Explain the meaning of the sentence, "In a random process, the trials are memoryless."
- Trials do not recall what has happened in the past and used them to make changes on
what's going to happen in the future
In probability, what is an experiment? - In probability, an experiment is any process with
uncertain results that can be repeated. The result of any single trial of the experiment is
not known ahead of time. However, the results of the experiment over many trials
produce regular patterns that allow accurate predictions.
A(n) _______ is any collection of outcomes from a probability experiment. - event
What is a probability model? - A probability model lists the possible outcomes of a
probability experiment and each outcome's probability. A probability model must satisfy
Rules 1 and 2 of the rules of probabilities.
What is an unusual event? What cutoff points do statisticians typically use for identifying
unusual events? - An unusual event is an event that has a low probability of occurring.
Typically, an event with a probability less than 0.05 (or 5%) is considered unusual, but
this cutoff point is not set in stone. Statisticians typically use cutoff points of 0.01,0.05,
and 0.10
List the three methods in this section for determining the probability of an event. - 1. the
Empirical Method
2. the Classical Method
3. the Subjective Method
Surveys are probability experiments. Why? - Each time a survey is conducted, a
different random sample of individuals is selected. Therefore, the results of a survey are
likely to be different each time the survey is conducted because different people are
included.
The _________ method gives an approximate probability of an event by conducting a
probability experiment. The ____________ method of computing probabilities does not
require that a probability experiment actually be performed, rather it relies on counting
techniques. - empirical; classical
What requirement must be met in order to compute probabilities using the classical
method? - The classical method requires equally likely outcomes. An experiment has
equally likely outcomes when each outcome has the same probability of occurring
As the number of trials of an experiment increase, how does the empirical probability of
an event occurring compare to the classical probability of that event occurring? - The
, empirical probability will get closer to the classical probability as the number of trials of
the experiment increases due to the Law of Large Numbers. If the two probabilities do
not get closer, we may suspect that the dice are not fair.
In ______________, each individual has the same chance of being selected. Therefore,
we can use the classical method to compute the probability of obtaining a specific
sample - simple random sampling
What is a subjective probability? Explain why subjective probabilities are used. - A
subjective probability is a probability that is determined based on personal judgement.
Subjective probabilities are legitimate and are often the only method of assigning
likelihood to an outcome. For instance, a financial reporter may ask an economist about
the likelihood of the economy falling into recession next year. Again, we cannot conduct
an experiment n times to find a relative frequency. The economist must use knowledge
of the current conditions of the economy and make an educated guess about the
likelihood of recession.
Explain the Law of Large Numbers. How does this apply to gambling casinos? - As the
number of repetitions of a probability experiment increases, the proportion with which a
certain outcome is observed gets closer to the probability of the outcome. This applies
to casinos because they are able to make a profit in the long run because they have a
small statistical advantage in each game.
What does it mean for two events to be disjoint? - Two events are disjoint if they have
no outcomes in common. Another name for disjoint events is mutually exclusive events.
If two events are mutually exclusive, it means that they cannot occur at the same time.
In a Venn diagram, what does the rectangle represent? What does a circle represent? -
The rectangle represents the sample space and each circle represents an event
How can you tell from a Venn diagram that two events are not disjoint? - The circles are
drawn to overlap.
State the Addition Rule for Disjoint Events. - If E and F are disjoint (or mutually
exclusive) events, then P(E or F) = P(E) + P(F)
For disjoint events E and F, how is P(E or F) related to P(E) and P(F)? - You add the
two probabilities together so P(E or F) = P(E) + P(F)
State the General Addition Rule. - For any two events, E and R, P(E or F) = P(E) + P(F)
- P(E and F)
Explain why we subtract P(E and F) when using the General Addition Rule. - We
subtract P(E and F) in order to avoid double counting. We are subtracting the probability
corresponding to the overlapping region E and F
