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47 Cards in this Set

  • Front
  • Back

Indicate if the following represents independent events. Explain briefly.




The gender of the consumer at the ATM?

Independent , because the outcome of one trial doesn't influence or change the outcome of another.

In the long run, outcomes settle into patterns that are consistent and predictable is called _____________________.
Random phenomena
Even though commercial airlines have excellent safety records, in the weeks following a crash, airlines often report a drop in the number of passengers, probably because people are afraid to risk flying.



A) A travel agent suggests that since the Law of Averages makes it highly unlikely to have two plane crashes within a few weeks of each other, flying soon after a crash is the safest time. What do you think?




B) If the airline industry proudly announces that it has set a new record for the longest period of safe flights, would you be reluctant to fly? Are the airlines due to have a crash?

A.) There is no such thing as the​ "Law of​ Averages." The overall probability of an airplane crash does not change due to recent crashes.



B.) There is no such thing as the​ "Law of​ Averages." The overall probability of an airplane crash does not change due to a period in which there were no crashes.





Each observation or experimental run is called a ___________.




Ex: 1. Tossing a coin four times


2. Rolling a Dice


3. Pulling 3 red marbles out of a bag of ten, 4 of which are red and 6 that are blue.



Trial

______________ is the result of a trial.




Ex: If I flipped a coin and it landed on heads, the _________ would be heads.

Outcome

Outcomes or combination of outcomes is called an ________________.




Ex:


- Getting a Tail when tossing a coin


- Rolling a "5".




An event can include several outcomes:


- Choosing a "King" from a deck of cards (any of the 4 Kings)


- Rolling an "even number" (2, 4 or 6)

Event

________________ is the "long run likelihood” or long run relativefrequency and can be written as a fraction, decimal, or percentage.




Ex: 50% means 50 per 100(50% of this box is green)

Probability
_____________ is the outcome of one trial doesn’t influence or change the outcome of another.



Ex: If you toss a coin fifty times, each coin toss is an ______________.



Independent trials

The collection of all possible outcomes is called _______________.




Ex:


Experiment 1:


- What is the probability of each outcome when a dime is tossed?


- Outcomes: The outcomes of this experiment are head and tail.


- Probabilities: P(head) = 1/2 or P(tail) = 1/2

Sample space

The long-run relative frequencyof repeated, independent events eventually produces the true relative frequency as the number of trials increases is called __________________.




Ex: If a fair coin (where heads and tails come up equally often) is tossed 1,000,000 times, about half of the tosses will come up heads, and half will come up tails. The heads-to-tails ratio will be extremely close to 1:1. However, if the same coin is tossed only 10 times, the ratio will likely not be 1:1, and in fact might come out far different, say 3:7 or even 0:10.

Law of Large Numbers

____________________ of an event is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials).




Ex: A survey was conducted to determine students' favorite breeds of dogs. Each student chose only one breed.What is the probability that a student's favorite dog breed is Lab out of six other dogs?

Empirical probability

_________________ is things have to even out in the short run.This is NOT THE WAY random phenomena work and it can lead to bad business decisions.



Ex:


• Is a good hitter in baseball who has struck out the last six times due for a hit his next time up?


• If the stock market has been down for the last 3 sessions, is it due to increase today?

“Law of Averages”

_____________, __________________, and ________________ are the three types of probabilty.

Empirical, Theoretical, and Subjective

________________ of an event is the number of ways that the event can occur, divided by the total number of outcomes. It is finding the probability of events that come from a sample space of known equally likely outcomes.




Ex: Find the probability of rolling a six on a fair die.


Answer: The sample space for rolling is die is 6 equally likely results: {1, 2, 3, 4, 5, 6}.The probability of rolling a 6 is 1/6.

Theoretical probability

_________________ is a probability derived from an individual's personal judgment about whether a specific outcome is likely to occur. It contains no formal calculations and only reflects the subject's opinions and past experience.




Ex: You think you have an 80% chance of your best friend calling today, because her car broke down yesterday and she’ll probably need a ride.

Subjective Probability

Two events are _______________ or _____________ if they cannot occur at the same time.




Note: Rule #5 of Probability

Mutually exclusive or disjoint

The probability is positive and less than or equal to 1.




0 ≤ p(A) ≤ 1

Properties of Probabilities

The probability of an event needs to be positive between 0 and 1. For example, the probability of getting heads are .50 and tails are .50. These fall between 0 and 1, and this is a _________________________.

Property of Probabilities

The probability of the sure event is 1.




p(S) = 1




This is ______________

Properties of Probability

The sum of getting heads is always equal to 1 is considered one of the ________________.

Properties of Probability
The sum of the probabilities of an event and its complementary is 1, so the probability of the complementary event is:



P(A-hat) = 1 - P(A)




This is ______________ rule of probability.

Rule 3: Complement Rule

- Suppose you know that the probability of getting the flu this winter is 0.43. What is the probability that you will not get the flu?




-Suppose that we flip eight fair coins. What is the probability that we have at least one head showing?




This is an example of rule ____________________.




Answer these.

Rule 3: Complement Rule




1 - 0.43 = .57




1 - (1/2)^8 = .996

When two events, A and B, are independent, the probability of both occurring is:




P(A and B) = P(A) · P(B)




This is ______________ rule of probability.

Rule 4: Multiplication Rule for Independent Events (AND)

- A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die.




- A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and then an eight?




- A jar contains 3 red, 5 green, 2 blue and 6 yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and then a yellow marble?




The probability rule being used is ________________. Solve Problems.

1/2 * 1/6 = 1/12




4/52 * 4/52 = 16/2704




5/16 * 6/16 = 30/256




Rule 4: Multiplication Rule for Independent Events (AND)

If A and B are mutually exclusive, then:




p(A unión B) = p(A) + p(B)




This is ______________ rule of probability.



Rule 5: Addition Rule for Mutually Exclusive (OR)

- A spinner has 4 equal sectors colored yellow, blue, green, and red. What is the probability of landing on red or blue after spinning this spinner?




- A glass jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If a single marble is chosen at random from the jar, what is the probability that it is yellow or green?




- A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?



The rule being used is ________________

1/4 + 1/4 = 1/2




4/10 + 3/10 = 7/10




1/6 + 1/6 = 1/3




Rule 5: Addition Rule for Mutually Exclusive (OR)

When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is:



P(A or B) = P(A) + P(B) - P(A and B)




This is ______________ rule of probability.

General Addition Rule

- In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?




- On New Year's Eve, the probability of a person having a car accident is 0.09. The probability of a person driving while intoxicated is 0.32 and probability of a person having a car accident while intoxicated is 0.15. What is the probability of a person driving while intoxicated or having a car accident?




These are examples of the __________________ rule.


13/30 + 9/30 - (5/30) = .567




.09 + .32 - .15 = .26




Rule 6: General Addition Rule

When two events, A and B, are dependent, the probability of both occurring is:



P(B|A) = P(A∩B) / P(B)


or


P(B|A) = P(A∩B) / P(A).




This is ______________ rule of probability.

Rule 7: Conditional Probability Rule

- A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed the second test?




- A jar contains black and white marbles. Two marbles are chosen without replacement. The probability of selecting a black marble and then a white marble is 0.34, and the probability of selecting a black marble on the first draw is 0.47. What is the probability of selecting a white marble on the second draw, given that the first marble drawn was black?




- The probability that it is Friday and that a student is absent is 0.03. Since there are 5 school days in a week, the probability that it is Friday is 0.2. What is the probability that a student is absent given that today is Friday?




- At Kennedy Middle School, the probability that a student takes Technology and Spanish is 0.087. The probability that a student takes Technology is 0.68. What is the probability that a student takes Spanish given that the student is taking Technology?




These are examples of the ________________ rule.

P (Second|First) = P(First and Second) / P(First)


.25/.42 = .60




0.34/0.47 = 0.72




.03/.2 = .15




.087/.68 = .13




Conditional Probability Rule

The probability that two events E and F both occur is:



P(E and F) = P(E) • P(F|E)




This is ______________ rule of probability.

General Multiplication Rule

- An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?






These are examples of the ____________ rule.

P(A ∩ B) = (4/10) * (3/9) = 12/90 = 2/15




General Multiplication Rule

The probabilities that an adult man has high blood pressure​ and/or high cholesterol are shown in the table. Are high blood pressure and high cholesterol​ independent? Explain
The probabilities that an adult man has high blood pressure​ and/or high cholesterol are shown in the table. Are high blood pressure and high cholesterol​ independent? Explain




No, because the outcome of one influences the probability of the other.
The word​ "or" in probability implies that we should use the _____________________.

Addition Rule

Suppose that in a certain community, the probability of a randomly selected individual having red hair is 0.08 and the probability of a randomly selected individual being left-handed is 0.15. What additional information would be needed to find the probability of randomly selecting an individual in this community who has red hair or is left-handed?
We would need to know the percentage of individuals in the community who have red hair and are​ left-handed.
Which probability method requires that the probability experiment be performed and uses the results to estimate the probability of a particular outcome?
Empirical​ (relative frequency)
The Addition Rule P(E or F) = P(E)+ P(F) only applies to _____________ events.
Disjoint
The Multiplication Rule P(E and F) = P(E)• P(F) only applies to ________________ events.
Independent
P (Selecting a Women) = 251/478 or .525 would be an example of ______________ probabilty.

P (Selecting a Women) = 251/478 or .525 would be an example of ______________ probabilty.

Marginal Probability

P (Selecting a Women who prefers Cameras) = 91/478 or .190 would be an example of _______________ probabilty.

P (Selecting a Women who prefers Cameras) = 91/478 or .190 would be an example of _______________ probabilty.

Joint Probability

Fifteen percent of employees in a company have managerial position, and 25 percent of the employees in the company have MBA degrees. Also, 60 percent of the managers have MBA degrees.

A. Find the proportion of employees who are managers and have MBA degrees.


B. Find the proportion of MBAs who are managers.

P(Manager ∩ MBA) = P(MBA | Manger) * P(Manger)

= 0.60 * 0.15


= 0.09




P( Manger | MBA) = P( Manger ∩ MBA) / P(MBA) = 0.09 / 0.25


= 0.36

_____________ are a picture to help think through thedecision-making process.



Example: Defects will always occur, so a quality engineer in charge of the production process must monitor the number of defects and take action if the process seems out of control.

Tree Diagrams

________________ is a table showing the distribution of one variable in rows and another in columns, used to study the association between the two variables.

Contingency table
Two events that do not occur at the same time, also known as mutually exclusive events are called ________________.



Ex: Being a freshman and being a sophomore would be considered_____________ because an individual cannot be classified as both at the same time.

Disjoint Events

Unrelated events that are not disjoint; the outcome of one event does not impact the outcome of the other event is called ___________________.



Ex: If there is no relationship between gender and class status, then we could say that being a man and being a senior are _______________; there are some men who are seniors thus they are not disjoint events.

Independent Events

What do three things needed to be checked to test if an event is independent?

1. P(A)×P(B)=P(A∩B)


2. P(A∣B)=P(A)


3. P(B∣A)=P(B)

What do you need to do to check if events are disjoint?

If two events are disjoint, then the probability of them both occurring at the same time is 0.

Disjoint: P(A and B) = 0