Statistics I EXAM: Chapters 4-5 (Based on Possible Open Ended Questions)

An experiment is an action where an outcome CANNOT be predicted with certainty. An event is a specified result that may or may not occur when experiment is performed. An example of an experiment would be like a coin toss. The events in this experiment would be whether the coin lands on heads or tails.

Roughly speaking, what is an experiment? An event? Can you give examples of both?

This is when events have the same chance of happening (this is like when we roll a dice or when the coin lands on heads or tails)

Considering the equal-likelihood model of probability, what is it?

This can be found by having the number of ways an event can occur over the total number of outcomes possible.

When we consider the equal-likelihood model of probability, how can you find the probability of an event?

In terms of selecting from either population or sample, there is no difference between selecting a member at random from a population or a sample. In general, you would need to choose exactly one member from each group with equal probability.

What is the difference between selecting a member at random from a finite population and taking a simple random sample of size 1?

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Let’s say an experiment has 20 possible outcomes, all equally likely. An event can occur in five ways. The probability that the event occurs is ____________.

Venn Diagrams

What types of graphical displays are useful for portraying events and relationships among events?

1) Sum the row totals.

2) Sum the Column Totals

3) Sum the frequencies within each cell from the table.

Identify three ways in which the total number of observations of bivariate data can be obtained from the frequencies in a contingency table.

This is the probability that event B occurs given that event A occurs

What is conditional probability?

This event would’ve been assumed to be an event that already occurred.

What event would be considered the “given event.”

An example would be when you flip a coin twice and you would have to calculate the probability of getting heads on the second outcome, GIVEN that on the first flip you got tails.

Give an example of the conditional probability of an event being the same as the unconditional probability of the event.

The general multiplication rule equals P(A|B) x P(B) while conditional probability rule equals P(A&B)/P(B)

Can you state the 2 rules regarding the general multiplication rule and the conditional probability rule?

The general multiplication rule and the conditional probability rule are like variations of each other. The general multiplication rule requires the conditional probability answer to find the result for P(A&B) while the conditional probability rule requires the same things.

What is the relationship between general multiplication and conditional probability?

The general multiplication and conditional probability are closely related, but they all have their differences- the general multiplication rule allows you to compute the joint probability of 2 events occurring together (for both independent and dependent events). Conditional Probability is when you want to find the probability of one event given the other - joint occurrence of events.

Why are 2 different variations of essentially the same rule emphasized?

This means that outcome of event A does not affect or change the outcome of event B.

What does it mean for event B to be independent of event A?

The joint probability can be obtained by multiplying their marginal probabilities (we call this the special multiplication rule)

If event A and event B are independent, how can their joint probability be obtained from their marginal probabilities?

When 4 events are exhaustive, this means that at least one of those four events must occur.

What does it mean for 4 events to be exhaustive?

If 4 events are mutually exclusive, this means that the 4 events do NOT have outcomes in common. → No 2 events can occur at the same time (non-overlapping)

What does it mean for four events to be mutually exclusive?

Exhaustive events are not necessarily mutually exclusive- exhaustive events mean that at least one of these events must occur. Mutually exclusive events aren’t necessarily exhaustive because they might not cover all possible outcomes - they might not account for them.

Are exhaustive events necessarily mutually exclusive? Are mutually exclusive events necessarily exhaustive? Explain your answers.

A complement of an event would contain all outcomes that are NOT in the original event, which means that they cannot occur simultaneously. The complement would also encompass all possible outcomes within the sample space (definition for exhaustive events)

Explain why an event and its complement are always mutually exclusive and exhaustive.

Counting rules are statistical techniques used to determine the number of possible outcomes in various situations - especially when dealing with large sets or complex arrangements. These rules are crucial for calculating probabilities, analyzing combinations, permutations, etc.

What are counting rules? Why are they important?

The BCR often involves multiplying the number of outcomes or probabilities given to find the total number of possible outcomes.

Why is the Basic Counting Rule often referred to as the multiplication rule?

Events that are independent just means that their outcomes won’t affect one another - As long as both events don’t occur simuantanesly, they can still be independent and not mutually exclusive.

Even when 2 events are independent of each other, does that mean that they are mutually exclusive events?

This is when the arrangement of a set of events where the order does MATTER. The events must be arranged in a specific sequence.

What is a permutation? Define the meaning.

This is where the arrangement of a set of events doesn’t rely on order- the events do not have to be arranged in a specific sequence.

What is combination? Define the meaning.

The notation {X=3} represents a SPECIFIC event- which refers to the set of outcomes in which the random variable X takes the value 3. P(X=3) just represents the probability of the random variable X equaling 3.

Let X denote the number of siblings of a randomly selected student. Explain the difference between {X=3} and P(X=3).

The mean of a discrete random variable generalize the average or EXPECTED value of a set of values.

What concept does the mean of a discrete random variable generalize?

Trial

In probability and statistics, what is each repetition of an experiment called?

1. The experiment (each trial) has two possible outcomes, denoted generically s, for success, and f, for failure.

2. The trials are independent, meaning that the outcome on one trial in no way affects the outcome on other trials.

3. The probability of a success, called the success probability and denoted p, remains the same from trial to trial.

What are the 3 conditions are repeated trials of an experiment called Bernoulli trials?

Binomial coefficients are significant to Bernoulli trials, as they provide a way to calculate the number of successful outcomes in a fixed number of independent trials.

What is the significance of binomial coefficients with respect to Bernoulli trials?

Although Binomial probability tables can offer the advantage of readily providing probabilities for various binomial distributions whitout requiring complex calculations - but they are limited to specific combinations of sample size and probability success.

Discuss the pros and cons of binomial probability tables.

A common probability distribution that models the probability of obtaining one of two outcomes under a given number of parameters.

What is the binomial distribution?