STA2014 - Chapter 5 : Probability

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

Addition

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.

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

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.

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.

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

0

disjoint event

events that have no outcomes in common

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

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.

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.

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.

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​}.

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.

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.

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.

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.

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.

never

​P(never)<0.05.

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.

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