Math 1040 Chapter 5

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Math 1040 Chapter 5
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2015-03-11 19:26:35
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A list of principles and vocabulary from Chapter 5
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  1. T or F

    Probability forms the basis of inferential statistics.
    True
  2. T or F

    Probability measures the likelihood of an outcome that has already occurred.
    False, that has not yet occurred
  3. What is 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.
  4. What is an outcome (or simple event)?
    The result of one trial of a probability experiment.
  5. What is sample space?
    The list of all possible results of a probability experiment.
  6. What is an event?
    Any collection of outcomes from a probability experiment. An event consists of one or more outcomes.

    Events with one outcome are called simple events.

    In general, events are denoted with capital letters, like "E".
  7. In probability, what is an experiment?
    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.
  8. State Rules 1 and 2 of the rules of probabilities:
    (1) The probability of any event, E, P(E), must be greater than or equal to "zero" and less than or equal to 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.
  9. What is a probability model?
    A probability model lists the possible outcomes of a probability experiment and each outcome's probability. A probability model must satisfy Rules 1 and 2 of the rules of probabilities.
  10. If an event is impossible...
    the probability of the event is 0.
  11. If an event is a certainty,...
    the probability of the event is 1.
  12. The closer a probability is to 1,...
    the more likely the event will occur.

    An event with probability of 0.8 is more likely to occur than an event with probability 0.75
  13. The closer a probability is to 0,...
    the less likely the event will occur.
  14. T or F

    An event with probability 0.75 *must occur 75 times out of 100.
    False, *does not have to occur 75 times out of 100. We expect this 0.75 (roughly) to occur.
  15. What is an unusual event? What cutoff point do statisticians typically use for identifying unusual events?
    An event that has a low probability of occurring. This is usually an event with a probability of less than 0.05, but this is not concrete.

    The researcher and the context of the problem determine the probability that separates unusual events from not so unusual events.
  16. Explain how to approximate probabilities using the empirical approach:
    The probability of an event, E, occurring is approximately the # of times an event, E, is observed divided by the # of repetitions (or trials) of the experiment.

    P(E)  relative frequency of E, or



    The result from this method is always approximate because different trials of the experiment lead to different outcomes and, therefore, different estimates of P(E).
  17. What are surveys?
    Surveys are probability experiments. Each time a survey is conducted, a different random sample of individuals is selected. Therefore, the results of a survey are likely to be different each time the survey is conducted because different people are included.
  18. What requirement must be met in order to compute probabilities using the classical method? What is the formula used for calculating probabilities using the classical method?
    The classical method requires equally likely outcomes. It does not require that an probability experiment actually be performed and does relies on counting techniques.

    If an experiment has "n" equally likely outcomes and if the # of ways that an event, E, can occur is "m", then the probability of E, P(E), is



    So, if "S" is the sample space of this experiment, then



    where N(E) is the # of outcomes in E, and N(S) is the # of outcomes in the sample space.
  19. As the # of trials of an experiment increase, the empirical probability will get...
    closer to the classical probability. This is according to the rule of large numbers. It is possible that the two probabilities differ because they have unequally likely events.

    If the two probabilities do not get closer, we may suspect that the dice, or whatever is being used to produce outcomes, is not fair.
  20. What is subjective probability?
    A probability that is determined based on personal judgment. These are legitimate and are often the only method of assigning likelihood to an outcome.
  21. 5.2

    Define disjoint (mutually exclusive).
    two events are disjoint if they have no outcomes in common. For example, the probability of drawing a king or drawing a queen from a deck of 52 cards are two disjoint events. If the context becomes drawing a king or a queen and drawing a spade from a deck of 52 cards, the events are no longer disjoint.
  22. 5.2

    In a Venn Diagram, the _________ represents the ______ _____. Each ______ represents a(n) _____.
    rectangle, sample space, circle, event
  23. 5.2

    You can tell from a Venn diagram that two events are not disjoint (mutually exclusive) when the ______ __ ___ _______ any ______ ________.
    events do not contain, common outcomes
  24. 5.2

    For disjoint events, P(E or F) (is/is not) related to (P(E)/P(F)/both P(E) and P(F)) when (one/both/neither) outcome totals are (the same/different). P(E or F) is (the same as/different than) adding the outcomes of (P(E)/P(F)/both P(E) and P(F)) together.
    is, both P(E) and P(F), both, the same, the same as, both P(E) and P(F)
  25. 5.2

    What is the Addition Rule for Disjoint Events?
    If E and F are disjoint (mutually exclusive) events, then P(E or F) = P(E) + P(F)

    This can be extended to more than two disjoint events.

    In general, if E, F, G,... each have no outcomes in common (they are pairwise disjoint), then P(E ororor...)=P(E) + P(F) + P(G) + ...
  26. 5.2

    State the General Addition Rule:
    For any two events E and F, P(E or F)=P(E) + P(F) - P(E and F)

    This is because if P(E and F) are not subtracted, then these values are double-counted when adding these outcomes and considering all outcomes as equally likely to occur.
  27. 5.2

    Define the complement of an event E. What is the formula? State the Complement Rule.
    Denoted EC, this is all outcomes in the sample space "S" that are not outcomes in the event E.

    P(E or EC) = P(E) + P(EC) = P(S) = 1

    Subtract P(E) from both sides and the complement rule is obtained:

    P(EC) = 1 - P(E)

    If E represents any event and EC represents the complement of E, then P(EC) = the area outside the circle in a Venn Diagram.

    This is used when finding the probability of two or more events when the phrase "at least" is involved.

    For example, you draw two cards from a deck and want to know what is the probability that at least one is a king, you can subtract the probability that none are kings from one (Complement Rule).

    Another example: You receive a shipment of six TVs. Two are defective. If two TVs are randomly selected, what is the probability that both work? That at least one of the two does not work (hint: this is the same as finding out the probability that at least one does work)?


    P(both work) = .400

    P(at least one of the two does not work) = .600
  28. 5.3

    What are independent events? Dependent events?
    Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F.

    Two events E and F are dependent if the occurrence of event E in a probability experiment affects the probability of event F.
  29. 5.3

    Why are events "draw a heart" and "roll an even number" independent events?
    Each event has no impact on the probability of the other (the results of the other).
  30. 5.3

    Why are events "woman 1 survives the year" and "woman 2 survives the year" dependent if the two women live in the same complex?
    Because there will likely be circumstances in the complex where the well-being of one woman can or will impact the well-being of the other woman.
  31. 5.3

    When we take a ____ _____ sample from a _____ _____ ______ population, we make the __________ of ____________ even though the ______ are technically _________.

    The general rule of thumb for assuming independence is if the ______ ____ "n" is __ ____ ____ ___ of the __________ size N (n0.05N), __ ______ ____________.
    very small, very large finite, assumption, independence, events, dependent, sample size, no more than 50%, population, we assume independence.
  32. 5.3

    T or F

    Disjoint events are independent.
    False, knowing that two events are disjoint (mutually exclusive) means that the events are not independent.
  33. 5.3

    Give the Multiplication Rule for Independent Events:
    If E and F are independent events, then P(E and F) = P(E) x P(F)

    If E1, E2, E3,...,En are independent events, then P(E1 and E2 and E3 and ... and En) = P(E1) x P(E2) x ... x P(En)

    For example, find the probability of correctly answering the first 3 questions on a multiple choice test if random guesses are made and each question has 6 possible answers.

    (1/6)*(1/6)*(1/6)=0.004630

    • If these events were not independent, this equation would not be used.
  34. 5.3

    T or F

    The phrase at least means "greater than or equal to".
    True. Usually when computing probabilities involving the phrase "at least", use the Complement Rule: P(EC) = 1 - P(E)
  35. 5.3

    Rules of Probability: Give all five rules of probability.
    Rule 1: The probability of any event must be between 0 and 1, inclusive. If we let E denote any event, the 0  P(E)  1.

    Rule 2: The sum of the probabilities of all outcomes in the sample space must equal 1. That is, if the sample space S = {e1, e2,...,e}, then

    • P(e1) + P(e2) + ... + P(en) = 1
    • Rule 3: If E and F are disjoint events, then P(E or F) = P(E) + P(F).

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

    Rule 4: If E represents any event and EC represents the complement of E, then P(EC) = 1 - P(E).

    Rule 5: If E and F are independent events, then P(E and F) = P(E) x P(F)

    or probabilities use the Addition Rule.

    or probabilities imply addition.

    and probabilities use the Multiplication Rule.

    and
    probabilities imply multiplication.
  36. 5.4

    What does the notation  represent?
    Conditional probability, "the probability of event F given event (outcome) E." One outcome does effect the likelihood of the other. RULES USED FOR CONDITIONAL PROBABILITY ARE DIFFERENT THAN THE RULES USED WHEN ONE OUTCOME DOES NOT EFFECT THE LIKELIHOOD OF THE NEXT.

    It is the conditional probability that event F occurs given that event E has occurred.

    S = {1, 2, 3, 4, 5, 6}

    Probability that a 3 is rolled is 1 in 6. If we are told the outcome of roll two is an odd number, P(3 is rolled given roll is odd) = 1 in 3 since S = {1, 3, 5}
  37. 5.4

    What is the Conditional Probability Rule?
    If E and F are any two events, then...



    The probability of event F occurring, given the occurrence of event E is found by dividing the probability of E and F by the probability of E. It is also found by dividing the number of outcomes in E and F by the number of outcomes in E.
  38. 5.4

    The probability that two events E and F both occur is...[formula]


    This can also be done using a tree.
  39. 5.4

    State the definition for independence using conditional probabilities:
    Two events E and F are independent if

  40. 5.5 (counting techniques that can be used along with the classical method)

    State the multiplication rule of counting:
    if a task consists of a sequence of choices in which there are "p" selections for the first choice, "q" selections for the second choice, "r" selections for the third choice, and so on, the the task of making these selections can be done in...

     different ways
  41. 5.5 (counting techniques that can be used along with the classical method)

    Define n factorial.
    n! If  is an integer, the factorial symbol, n!, is defined as...



    0! = 1

    1! = 1
  42. 5.5 (counting techniques that can be used along with the classical method)

    Define permutation and give the formula.
    an ordered arrangement in which "r" objects are chosen so that  and repetition is not allowed. the symbol



    represents the number of permutations of "r" objects selected from "n" objects.
  43. 5.5 (counting techniques that can be used along with the classical method)

    Give the formula for the number of permutations of "n" distinct objects taken "r" at a time.



    • (1) The "n" objects are distinct,
    • (2) Repetition of objects is not allowed, and
    • (3) Order is important
  44. 5.5 (counting techniques that can be used along with the classical method)

    Define a combination.
    A combination is a collection in which "r" objects are chosen from "n" distinct objects without repetition and without regard to order. "r" is always less than or equal to "n". The symbol



    represents the number of combinations of "n" distinct objects taken "r" at a time.
  45. 5.5 (counting techniques that can be used along with the classical method)

    T or F

    The only difference between a permutation and a combination is that we disregard order in combinations.
    True
  46. 5.5 (counting techniques that can be used along with the classical method)

    Give the formula for the number of combinations of "n" distinct objects taken "r" at a time.



    • (1) The "n" objects are distinct,
    • (2) Repetition of objects is not allowed, and
    • (3) Order is not important
  47. 5.5 (counting techniques that can be used along with the classical method)

    Sometimes we want to arrange objects in order but some of the objects are not distinguishable; that is, for permutations with ___________ items, the number of ____________ of n objects of which __ are of one kind, __ are of a second kind, …, and __ are of a kth kind is given by

    [give formula]
    nondistinct, permutations, n1, n2, nk,

  48. 5.5 (counting techniques that can be used along with the classical method)

    T or F

    The counting techniques presented in section 5.5 can be used along with the classical method to compute certain probabilities.
    True
  49. 5.5

    Give the four types of Combinations and Permutations and their formulas.
    Combination:

    The selection of r objects from a set of n different objects when the order the objects are selected in does not matter (AB is the same as BA) and there is no repetition (an object cannot be selected more than once).



    Permutation of Distinct Items WITH Replacement:

    Selection of "r" objects from a set of "n" different objects when order does matter (AB is not the same as BA) and repetition is allowed (an object may be selected more than once).



    Permutation of Distinct Items WITHOUT Replacement:

    selection of "r" objects from a set of "n" different objects when order does matter and repetition is not allowed.



    Permutation of Nondistinct Items WITHOUT Replacement:

    Number of ways "n" objects can be arranged when there are  of one kind,  of a second kind,..., and  of kth kind. In these cases, 

  50. 5.6

    Give the flowchart for Probability Rules:
  51. 5.6

    Give the flowchart that determines the appropriate Counting Technique to Use.


    When working with a sequence of choices, the counting techniques we use are:

    The Multiplication Rule if the number of choices at each stage is independent of previous choices.

    The tree diagram if the number of choices at each stage is not independent of previous choices.

    When working with the number of arrangements of items, we want to know "does the order of selection matter"? If it does, we want to know if we are arranging all the items available or just a short subset.
  52. Regarding replacement: when asked what the P of two events is WITHOUT replacement, two fractions are multiplied. When asked what the P of two events is WITH replacement, the P of one event can be squared to find the answer.

    Give an example of these principles.
    Two cards are randomly selected from a standard 52-card deck.

    Find P(1st and 2nd cards drawn are both spades without replacement)

    Find P(1st and 2nd cards drawn are both spades with replacement)





    Answers:

    Without replacement: 0.059 = (13/52)*(12/51)

    With replacement: 0.063 = (13/52)^2
  53. Determine the probability that at least 2 people in a room of 8 people share the same birthday, ignoring leap years and assuming each birthday is equally likely, by answering the following questions:

    (a) Compute the probability that 8 people have different birthdays.

    (b) The complement of "8 people have different birthdays" is "at least 2 share a birthday". Use this information to compute the probability that at least 2 people out of 8 share the same birthday.
    (a) 0.9257

    (b) 0.0743
  54. Among 19 to 27 year olds, 35% say they have used a computer while under the influence of alcohol. Suppose six 19 to 27 year olds are selected at random.

    (a) What is the probability that at least one has not used a computer while under the influence of alcohol?

    (b) What is the probability that at least one has used a computer while under the influence of alcohol?
    (a) The likelihood that all individuals randomly selected have used a computer while under the influence of alcohol is equal and the problem states "at least". The Complement Rule can be used for this problem.

    Find the P that all six individuals have used a computer while under the influence of alcohol, we find .35^6. This turns out to be .0018.

    Using the Complement Rule, P(at least one has not used a computer while under the influence of alcohol) = 1 - .0018. This equals .9981, or 99.81%.

    (b) The likelihood that all individuals randomly selected have used a computer while under the influence of alcohol again is equally likely to all participants and the problem as "at least". The Complement Rule can be used to help solve this problem as well.

    This time, we are looking for P(at least one has used a computer while under the influence of alcohol). This is 75% of the of 19 to 27 year olds, or .75. We raise .75 to the 4th power to find the likelihood that all 4 randomly selected individuals have not used a computer while under the influence of alcohol. This turns out to be .3164.

    Use the Complement Rule to answer part (b). P(at least one has used a computer while under the influence of alcohol) = 1 - .3164. The outcome is .6836, or 68.36%.

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