STA2014 - Chapter 5 : Probability

probability model rules

A probability model lists the possible outcomes of a probability experiment and each​ outcome's probability. It has two rules. 1. The probability of any event​ E, P(E), must be greater than or equal to 0 and less than or equal to one. 2. The sum of the probabilities of all outcomes must equal 1.

The word or in probability implies that we use the ________ Rule.

Addition

What does it mean for an event to be​ unusual? Why should the cutoff for identifying unusual events not always be​ 0.05?

An event is unusual if it has a low probability of occurring. The choice of a cutoff should consider the context of the problem.

Describe what an unusual event is. Should the same cutoff always be used to identify unusual​ events? Why or why​ not?

An event is unusual if it has a low probability of occurring. The same cutoff should not always be used to identify unusual events. Selecting a cutoff is subjective and should take into account the consequences of incorrectly identifying an event as unusual.

Explain the Law of Large Numbers. How does this law 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.

Determine whether the probabilities below are computed using the classical​ method, empirical​ method, or subjective method: The probability of having six girls in an six​-child family is 0.015625.

Classical method

Classical probability

Classical probability is used when each outcome in a sample space is equally likely.

Empirical probability

Empirical probability is based on observations obtained from probability experiments.

Determine if the following statement is true or false. When two events are​ disjoint, they are also independent.

False. The correct answer is False because two events are disjoint if they have no outcomes in common. In other​ words, the events are disjoint​ if, knowing that one of the events​ occurs, we know the other event did not occur. Independence means that one event occurring does not affect the probability of the other event occurring.​ Therefore, knowing two events are disjoint means that the events are not independent.

If E and F are not disjoint​ events, then​ P(E or ​F)=​________.

If E and F are not disjoint​ events, then​ P(E or ​F)= P(E) + P(F) - P(E and F)

The word and in probability implies that we use the​ ________ rule.

Multiplication

Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is​ impossible?

No. When a probability is based on an empirical​ experiment, a probability of zero does not mean that the event cannot occur. The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the​ experiment, as shown below. Just because the event is not​ observed, does not mean that the event is impossible.

If E and F are disjoint​ events, then P(E or F) =

P(E) + P(F).

If E and F are disjoint​ events, then P(E or F) =

P(E)+P(F)

Subjective probability

Subjective probability of an outcome is a probability obtained on the basis of personal judgment.

Describe the difference between classical and empirical probability.

The empirical method obtains an approximate empirical probability of an event by conducting a probability experiment. The classical method of computing probabilities does not require that a probability experiment actually be performed.​ Rather, it relies on counting​ techniques, and requires equally likely outcomes. The empirical method obtains an approximate probability of an even by conducting a probability experiment. The probability is approximate because different runs of the probability experiment lead to different​ outcomes, and,​ therefore, different estimates of the probability. The classical method of computing probabilities relies on counting​ techniques, and requires equally likely outcomes. An experiment has equally likely outcomes when each outcome has the same probability of occurring.

Bob is asked to construct a probability model for rolling a pair of fair dice. He lists the outcomes as​ 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, 11, 12. Because there are 11​ outcomes, he​ reasoned, the probability of rolling a three must be 1/11. What is wrong with​ Bob's reasoning?

The experiment does not have equally likely outcomes.

In a certain card​ game, the probability that a player is dealt a particular hand is 0.43. Explain what this probability means. If you play this card game 100​ times, will you be dealt this hand exactly 43 ​times? Why or why​ not?

The probability 0.43 means that approximately 43 out of every 100 dealt hands will be that particular hand.​ No, you will not be dealt this hand exactly 43 times since the probability refers to what is expected in the​ long-term, not​ short-term.

According to a certain​ country's department of​ education, 41.4​% of​ 3-year-olds are enrolled in day care. What is the probability that a randomly selected​ 3-year-old is enrolled in day​ care?

The probability that a randomly selected​ 3-year-old is enrolled in day care is .414.

Suppose you toss a coin 100 times and get 52 heads and 48 tails. Based on these​ results, what is the probability that the next flip results in a tail​?

The probability that the next flip results in a tail is approximately .48.

If a person spins a six-space spinner and then draws a playing card and checks its color​, describe the sample space of possible outcomes using 1, 2, 3, 4, 5, 6 for the spinner outcomes and B, R for the card outcomes.

The sample space is S = {1B,1R,2B,2R,3B,3R,4B,4R,5B,5R,6B,6R​}.

True or False​: In a probability​ model, the sum of the probabilities of all outcomes must equal 1.

True. In a probability​ model, the sum of the probabilities of all outcomes must equal 1.

Determine if the following statement is true or false. Probability is a measure of the likelihood of a random phenomenon or chance behavior.

True. The given statement is the definition of probability.

When a probability experiment is​ run, probabilities are approximated using the ________ approach.

empirical

​A(n) ________ is any collection of outcomes from a probability experiment.

event

In​ probability, a(n)​ ________ is any process that can be repeated in which the results are uncertain.

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 enable one to predict with remarkable accuracy.

Two events E and F are ________ if the occurrence of event E in a probability experiment does not affect the probability of event F.

independent

According to a center for disease​ control, the probability that a randomly selected person has hearing problems is 0.141. The probability that a randomly selected person has vision problems is 0.087. Can we compute the probability of randomly selecting a person who has hearing problems or vision problems by adding these​ probabilities? Why or why​ not?

​No, because hearing and vision problems are not mutually exclusive.​ So, some people have both hearing and vision problems. These people would be included twice in the probability.

never

​P(never)<0.05.

If n greater than or equals 0n≥0 is an​ integer, the factorial​ symbol, n!, is defined by the formulas below.

​n!=(n−1)•...•3•2•1 ​1!=1 ​0!=1

What is the probability of an event that is​ impossible?

0

disjoint event

events that have no outcomes in common