Chapter 5.1

Computing Probability Using the Classical Method

If an experiment has n equally likely outcomes and if the number of ways that an event E can occur is m, then the probability of E,P(E), is P(E)= (number of ways that E can occur)/ (number of possible outcomes) = m/n So, if S is the sample space of this experiment, then P(E)=N(E)/N(S) Where N(E) is the number of outcomes in E, and N(S) is the number of outcomes in the sample space.

Comparing Empirical Probabilities and Classical Probabilities

In comparing the results of Examples 7(a) and 7(b), notice that the two probabilities are slightly different. Empirical probabilities and classical probabilities often differ in value, but as the number of repetitions of a probability experiment increases, the empirical probability should get closer to the classical probability according to the Law of Large Numbers. However, it is possible that the two probabilities differ because having a boy and having a girl are not equally likely events. (Maybe the probability of having a boy is 0.505 and the probability of having a girl is 0.495.) If this is the case, then the empirical probability will not get closer to the classical probability because the events "boy" and "girl" are not equally likely.

Event

Is any collection of outcomes from a probability experiment. an event consists of one outcome or more than one outcome. E

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 more variability in short run experiment

Problem A pair of fair dice is rolled. Fair die are die where each outcome is equally likely. Approach To compute probabilities using the classical method, count the number of outcomes in the sample space and count the number of ways the event can occur. Then, divide the number of outcomes by the number of ways the event can occur.

Solution There are 36 equally likely outcomes in the sample space, as shown in Figure 1. So N(S)=36. The event E="roll a seven"={(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)} has six outcomes, so N(E)=6. Therefore, P(E)=P(roll a seven)= N(E)/N(S) =6/36 = 16 The probability of rolling a seven is 16.

Classical method

The classical method of computing probabilities does not require that a probability experiment actually be performed. Rather, it relies on counting techniques. The classical method of computing probabilities requires equally likely outcomes. An experiment has equally likely outcomes when each outcome has the same probability of occurring. Ex: when a fair die is thrown once, each of the six outcomes in the sample space, {1,2,3,4,5,6}, has an equal chance of occurring. Contrast this situation with a loaded die in which a five or six is twice as likely to occur as a one, two, three, or four.

A survey of 500 randomly selected high school students determined that 294 play organized sports. ​(a) What is the probability that a randomly selected high school student plays organized​ sports? ​(b) Interpret this probability.

(a) .588 (b) about 588

Suppose you physically simulate the random process of rolling a single die. ​(a) After 10 rolls of the​ die, you observe a​ "one" 3 times. What proportion of the rolls resulted in a​ "one"? ​(b) After 20 rolls of the​ die, you observe a​ "one" 2 times. What proportion of the rolls resulted in a​ "one"?

(a) 0.3 (b) 0.1

A bag of 100 tulip bulbs purchased from a nursery contains 30 red tulip​ bulbs, 35 yellow tulip​ bulbs, and 35 purple tulip bulbs. ​(a) What is the probability that a randomly selected tulip bulb is​ red? ​(b) What is the probability that a randomly selected tulip bulb is​ purple? ​(c) Interpret these two probabilities.

(a) 0.3 (b) 0.35 (c) 30 red, 35 purple

A baseball player hit 61 home runs in a season. Of the 61 home​ runs, 21 went to right​ field, 21 went to right center​ field, 9 went to center​ field, 9 went to left center​ field, and 1 went to left field. ​(a) What is the probability that a randomly selected home run was hit to right​ field? ​(b) What is the probability that a randomly selected home run was hit to left​ field? ​(c) Was it unusual for this player to hit a home run to left​ field? Explain.

(a) 0.344 (b) 0.016 (c) Yes, bc P(left field)<0.05

A survey of 100 randomly selected high school students determined that 70 play organized sports. ​(a) What is the probability that a randomly selected high school student plays organized​ sports? ​(b) Interpret this probability.

(a) 0.70 (b) 700

Determine whether the probabilities below are computed using the classical​ method, empirical​ method, or subjective method. Complete parts ​(a) through ​(d) below. (a) The probability of having eight girls in an eight​-child family is 0.00390625. (b) On the basis of a survey of 1000 families with eight ​children, the probability of a family having eight girls is 0.0064. (c) According to a sports​ analyst, the probability that a football team will win the next game is 0.48. (d) On the basis of clinical​ trials, the probability of efficacy of a new drug is 0.85.

(a) Classical method (b) Empirical method (c) Subjective method (d) Empirical method

Dominique, Roberto, Marco, Clarice, and John work for a publishing company. The company wants to send two employees to a statistics conference. To be​ fair, the company decides that the two individuals who get to attend will have their names randomly drawn from a hat. ​(a) Determine the sample space of the experiment. That​ is, list all possible simple random samples of size n=2. ​(b) What is the probability that Roberto and Marco attend the​ conference? ​(c) What is the probability that Clarice attends the​ conference? ​(d) What is the probability that John stays​ home?

(a) DR, DM, DC, DJ, RM, RC, RJ, MC, MJ, CJ (b) 0.1 (c) 0.4 (d) 0.6

Is the following a probability​ model?What do we call the outcome ​"blue"? (a) Is the table above an example of a probability​ model? (b) What do we call the outcome ​"blue​"?

(a) No​, because the probabilities do not sum to 1. (b) impossible event.

Key Concepts Regarding Probabilities

- If an event is impossible, the probability of the event is 0. - If an event is a certainty, the probability of the event is 1. - The closer a probability is to 1, the more likely the event will occur. - The closer a probability is to 0, the less likely the event will occur. - For example, an event with probability 0.8 is more likely to occur than an event with probability 0.75. - An event with probability 0.8 will occur about 80 times out of 100 repetitions of the experiment, whereas an event with probability 0.75 will occur about 75 times out of 100.

In a recent​ survey, it was found that the median income of families in country A was $57,900. What is the probability that a randomly selected family has an income less than $57,900​?

0.5

Methods for determining the probability of an event

1. The Empirical Method 2. The Classical Method 3. The Subjective Method

Simple random sampling

A sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equal chance of occurring.

Subjective probability

A subjective probability is a probability that is determined based on personal judgment.

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.

Unusual event

An unusual event is an event that has a low probability of occurring. Around 0.05

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.

Brad and Allison have three girls. Brad tells Allison that he would like one more child because they are due to have a boy. What do you think of​ Brad's logic?

Brad is incorrect due to the nonexistent Law of Averages. The fact that Brad and Allison had three girls in a row does not matter. The likelihood the next child will be a boy is about 0.5.

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.

Experiment

It 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.

Law of Averages

Not a law after all, memoryless property ex: having 4 girls and being "due" for a boy

Rules of Probabilities

P(E) means "the probability that event E occurs." 1. The probability of any event E, P(E), must be greater than or equal to 0 and less than or equal to 1. That is, 0≤P(E)≤1. 2. The sum of the probabilities of all outcomes must equal 1. That is, if the sample space S={e1,e2,⋯,en}, then P(e1)+P(e2)+⋯+P(en)=1

Let the sample space be S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E=​"an odd number less than 7​."

P(E)= 0.3

Empirical method

The empirical method gives an approximate probability of an event by conducting a probability experiment.

Long term proportion

The long-term proportion in which a certain outcome is observed is the probability of that outcome. So we say that the probability of observing a head is 12 or 50% or 0.5 because as we flip the coin more times, the proportion of heads tends to 0.5. This phenomenon is referred to as the Law of Large Numbers.

In a certain card​ game, the probability that a player is dealt a particular hand is 0.25. Explain what this probability means. If you play this card game 100​ times, will you be dealt this hand exactly 25 times? Why or why​ not?

The probability 0.25 means that approximately 25 out of every 100 dealt hands will be that particular hand.​ No, you will not be dealt this hand exactly 25 times since the probability refers to what is expected in the​ long-term, not​ short-term.

In a certain card​ game, the probability that a player is dealt a particular hand is 0.45. Explain what this probability means. If you play this card game 100​ times, will you be dealt this hand exactly 45 ​times? Why or why​ not?

The probability 0.45 means that approximately 45 out of every 100 dealt hands will be that particular hand.​ No, you will not be dealt this hand exactly 45 times since the probability refers to what is expected in the​ long-term, not​ short-term.

Computing Probability Using the Empirical Method

The probability of an event E occurring is approximately the number of times event E is observed divided by the number of repetitions (or trials) of the experiment. P(E) ≈ relative frequency ofE= (frequency of E/number of trials of experiment)

Suppose you toss a coin 100 times and get 59 heads and 41 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 0.41

If a person rolls a six-sided die 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 die outcomes and B, R for the card outcomes.​ (Make sure your answers reflect the order​ stated.)

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

If a person spins a six-space spinner and then flips a coin​, describe the sample space of possible outcomes using 1, 2, 3, 4, 5, 6 for the spinner outcomes and H, T for the coin outcomes

The sample space is S= {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6​}.

Sample space

The sample space, S, of a probability experiment is the collection of all possible outcomes for that experiment. S={1,2,3,4,5,6} of even numbers, E={2,4,6}

Why is the following not a probability​ model? Color, Probability Red, 0.2 Green, −0.4 Blue, 0.2 Brown, 0.4 Yellow, 0.3 Orange,0.3

This is not a probability model because at least one probability is less than 0.

Comparing Empirical Probabilities and Classical Probabilities

We just saw that the classical probability of rolling a seven is 16≈0.167. Suppose a pit boss at a casino rolls a pair of dice 100 times and obtains 15 sevens. From this empirical evidence, we would assign the probability of rolling a seven as 15100=0.15. If the dice are fair, we would expect the relative frequency of sevens to get closer to 0.167 as the number of rolls of the dice increases. In other words, 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 simple random sampling, 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.

Simulation

technique used to recreate a random event ex: Tactile; physically flipping a coin several times virtual; using a computer to pretend its flipping a coin

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 two must be 1/11. What is wrong with​ Bob's reasoning?

the experiment does not have equally likely outcomes

Random

unpredictable result or outcome