Probability (DP IB Analysis & Approaches (AA))

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Cards in this collection (41)

  • Define the term experiment in the context of probability.

    In probability, an experiment is a repeatable activity that has a result that can be observed or recorded.

  • What is a trial in the context of probability?

    In probability, a trial is one of the repeats of a probability experiment.

  • What is an outcome in the context of probability?

    In probability, an outcome is a possible result of a trial.

  • Define sample space.

    A sample space is the set of all possible outcomes of an experiment.

  • State the equation for the theoretical probability of event A occurring, in terms of n open parentheses A close parentheses and n open parentheses U close parentheses.

    The theoretical probability of event A occurring is straight P open parentheses A close parentheses equals fraction numerator n open parentheses A close parentheses over denominator n open parentheses U close parentheses end fraction

    Where:

    • straight P open parentheses A close parentheses is the probability of event A occurring

    • n open parentheses A close parentheses is the number of outcomes in the sample space that belong to event A

    • n open parentheses U close parentheses is the total number of possible outcomes in the sample space

    This equation assumes that all outcomes in the sample space are equally likely.

    The equation is in the exam formula booklet.

  • What does A to the power of apostrophe denote, in relation to an event A?

    A to the power of apostrophe is the complement of event A.

    This can be thought of as 'not A'. It is the event where A doesn't happen.

  • State the equation that connects the probabilities straight P open parentheses A close parentheses and straight P open parentheses A to the power of apostrophe close parentheses.

    The equation that connects the probabilities straight P open parentheses A close parentheses and straight P open parentheses A to the power of apostrophe close parentheses is the complementary events equation straight P open parentheses straight A close parentheses plus straight P open parentheses straight A to the power of apostrophe close parentheses equals 1

    Where:

    • straight P open parentheses A close parentheses is the probability of event A occurring

    • straight P open parentheses A to the power of apostrophe close parentheses is the probability of event A not occurring

    This equation is in the exam formula booklet.

  • True or False?

    The sum of the probabilities of all outcomes in a sample space can be less than 1.

    False.

    The sum of the probabilities of all outcomes in a sample space is equal to 1.

  • What is the intersection of two events?

    The intersection of two events is the event where both of the events occur.

    If the two events are A and B, then this is the event (A and B). It is denoted by A intersection B.

  • What is A union B the notation for?

    A union B is the notation for the union of events A and B.

    This is the event where A or B (or both) occurs.

  • State the equation that connects the probabilities straight P open parentheses A close parentheses, straight P open parentheses B close parentheses, straight P open parentheses A union B close parentheses and straight P open parentheses A intersection B close parentheses.

    The equation that connects the probabilities straight P open parentheses A close parentheses, straight P open parentheses B close parentheses, straight P open parentheses A union B close parentheses and straight P open parentheses A intersection B close parentheses is the combined events equation straight P open parentheses A union B close parentheses equals straight P open parentheses straight A close parentheses plus straight P open parentheses straight B close parentheses minus straight P open parentheses straight A intersection straight B close parentheses

    Where:

    • straight P open parentheses A close parentheses is the probability of event A occurring

    • straight P open parentheses B close parentheses is the probability of event B occurring

    • straight P open parentheses A union B close parentheses is the probability of event A or event B occurring

    • straight P open parentheses A intersection B close parentheses is the probability of event A and event B occurring

    This equation is in the exam formula booklet.

  • What are mutually exclusive events?

    Mutually exclusive events are events that cannot both occur at the same time.

    For example, 'roll a 1' and 'roll an even number' are mutually exclusive events, because they cannot both occur on the same roll of a dice.

  • What are independent events?

    Independent events are events where one occurring (or not) does not affect the probability of the other occurring.

  • True or False?

    If A and B are independent events, then P open parentheses A intersection B close parentheses equals straight P open parentheses A close parentheses plus straight P open parentheses B close parentheses.

    False.

    If A and B are independent events, then P open parentheses A intersection B close parentheses equals straight P open parentheses A close parentheses straight P open parentheses B close parentheses.
    I.e. multiply the probabilities to find the probability of both events occurring.

    If A and B are mutually exclusive events, then P open parentheses A union B close parentheses equals straight P open parentheses A close parentheses plus straight P open parentheses B close parentheses.
    I.e. add the probabilities to find the probability of one or the other (or both) events occurring.

    Both of those equations are in the exam formula booklet.

  • State the equation that connects the probabilities straight P open parentheses A close parentheses, straight P open parentheses B close parentheses and straight P open parentheses A union B close parentheses for mutually exclusive events.

    The equation that connects the probabilities straight P open parentheses A close parentheses, straight P open parentheses B close parentheses, straight P open parentheses A union B close parenthesesfor mutually exclusive events is straight P open parentheses A union B close parentheses equals straight P open parentheses A close parentheses plus straight P open parentheses B close parentheses.

    Where:

    • straight P open parentheses A close parentheses is the probability of event A occurring

    • straight P open parentheses B close parentheses is the probability of event B occurring

    • straight P open parentheses A union B close parentheses is the probability of event A or event B occurring

    This equation is in the exam formula booklet.

  • What is relative frequency?

    Relative frequency is the experimental probability of an outcome.

    It is calculated by dividing the frequency of the outcome (i.e. the number of times the outcome occurs) by the number of trials.

  • True or False?

    For two mutually exclusive events A and B, the probability straight P open parentheses A intersection B close parentheses can take on any value between 0 and 1.

    False.

    For two mutually exclusive events A and B, straight P open parentheses A intersection B close parentheses equals 0.

    This is because if A and B are mutually exclusive, they cannot both occur.

  • What is conditional probability?

    Conditional probability is where the probability of an event happening can vary depending on the outcome of another event.

  • What is straight P open parentheses A vertical line B close parentheses the notation for?

    straight P open parentheses A vertical line B close parentheses is the notation for the conditional probability of event A occurring, given that event B has occurred.

  • Define without replacement in the context of probability.

    In the context of probability, without replacement is when items are not returned to the set after being selected.

    This changes the probabilities for subsequent selections.

  • True or False?

    In conditional probability questions, the total number of items remains constant when selecting without replacement.

    False.

    In conditional probability questions, the total number of items changes when selecting without replacement.

  • True or False?

    The equation for the conditional probability of A given B is straight P open parentheses A vertical line B close parentheses equals fraction numerator straight P open parentheses A union B close parentheses over denominator straight P open parentheses A close parentheses end fraction.

    False.

    The equation for the conditional probability of A given B is straight P open parentheses A vertical line B close parentheses equals fraction numerator straight P open parentheses A intersection B close parentheses over denominator straight P open parentheses B close parentheses end fraction.

    This is given in the exam formula booklet.

  • True or False?

    Conditional probability can be calculated using sample space diagrams.

    True.

    Conditional probability can be calculated using sample space diagrams.

    For example, to find straight P open parentheses A vertical line B close parentheses using a sample space diagram:

    • reduce your sample space to just include outcomes for event B,

    • then find the proportion of those outcomes that also contains outcomes for event A.

  • What is the formula for straight P open parentheses A intersection B close parentheses in terms of P open parentheses A vertical line B close parenthesesand straight P open parentheses B close parentheses?

    A formula for straight P open parentheses A intersection B close parentheses can be written in the form straight P open parentheses A intersection B close parentheses equals straight P open parentheses A vertical line B close parentheses cross times straight P open parentheses B close parentheses.

    This is not in the exam formula booklet.

    But it can be found by rearranging straight P open parentheses A vertical line B close parentheses equals fraction numerator straight P open parentheses A intersection B close parentheses over denominator straight P open parentheses B close parentheses end fraction, which is in the formula booklet.

  • True or False?

    straight P open parentheses B intersection A close parentheses equals straight P open parentheses A intersection B close parentheses

    True.

    straight P open parentheses B intersection A close parentheses equals straight P open parentheses A intersection B close parentheses

    The events B intersection A (B and A both occur) and A intersection B (A and B both occur) are the same event, so their probabilities are the same.

  • True or False?

    straight P open parentheses B vertical line A close parentheses equals straight P open parentheses A vertical line B close parentheses

    False.

    In general P open parentheses B vertical line A close parentheses, the probability that B occurs given that A has occurred, is not equal to straight P open parentheses A vertical line B close parentheses, the probability that A occurs given that B has occurred.

  • What is the relationship between straight P open parentheses A vertical line B close parentheses and straight P open parentheses A close parentheses if A and B are independent?

    If A and B are independent, then straight P open parentheses A vertical line B close parentheses equals straight P open parentheses A close parentheses.

  • True or False?

    Conditional probabilities can be used to test whether two events are independent.

    True.

    If Aand B are two events then they are independent if straight P left parenthesis A vertical line B right parenthesis equals straight P left parenthesis A right parenthesis equals straight P left parenthesis A vertical line B apostrophe right parenthesis.

    I.e., if the probability of A happening does not depend on whether B happens or not, then the two events are independent.

  • What is a Venn diagram?

    A Venn diagram is a way to illustrate events from an experiment, and is particularly useful when there is an overlap between possible outcomes.

    A basic Venn diagram showing two overlapping sets labelled A and B
  • What does the rectangle in a Venn diagram represent?

    The rectangle in a Venn diagram represents the sample space (usually denoted by U).

  • True or False?

    In a Venn diagram, all the circles must overlap.

    False.

    In a Venn diagram, circles may overlap, depending on whether or not outcomes are shared between events, but they don't have to overlap.

    In particular, mutually exclusive events are represented by non-overlapping circles in a Venn diagram.

  • What does A intersection B represent in a Venn diagram?

    In a Venn diagram, A intersection B represents the region where the A and B circles overlap.

    A Venn diagram with two circles labelled A and B, and with the overlap between the two circles shaded in
  • What does A to the power of apostrophe represent in a Venn diagram?

    A to the power of apostrophe represents the regions that are not in the bold italic A circle in a Venn diagram.

    A Venn diagram with two circles labelled A and B, and with the parts not in the A circle shaded in
  • What does A union B represent in a Venn diagram?

    In a Venn diagram, A union B represents the regions that are in bold italic A or bold italic B or both.

    A Venn diagram with two circles labelled A and B, and with all the parts inside both circles shaded in
  • True or False?

    Independent events can be seen instantly in a Venn diagram.

    False.

    Independent events can not be seen instantly in a Venn diagram. You need to use probabilities to deduce if two events are independent.

  • True or False?

    In a Venn diagram showing frequencies, all the frequencies shown in the diagram should add up to the total frequency.

    True.

    In a Venn diagram showing frequencies, all the frequencies shown in the diagram should add up to the total frequency.

  • What is a tree diagram?

    A tree diagram is a way to show the outcomes of combined events.

    A tree diagram with two outcomes at the first set of branches, and two outcomes at each of the second set of branches.
  • True or False?

    The events on the branches of a tree diagram must be mutually exclusive.

    True.

    The events on the branches of a tree diagram must be mutually exclusive.

  • True or False?

    On a tree diagram, all the probabilities on a single set of branches should add up to 1.

    True.

    On a tree diagram, all the probabilities on a single set of branches should add up to 1.

  • How do you calculate the probability of two events happening together in a tree diagram?

    To find the probability that two events happen together in a tree diagram, you multiply the corresponding probabilities along their branches.

  • How do you find straight P open parentheses A union B close parentheses using a tree diagram?

    To find straight P open parentheses A union B close parentheses using a tree diagram, add together the probabilities of the combined outcomes that are part of that event:

    straight P open parentheses A union B close parentheses equals straight P open parentheses A intersection B close parentheses plus straight P open parentheses A intersection B to the power of apostrophe close parentheses plus straight P open parentheses A to the power of italic apostrophe union B close parentheses

    Alternatively, subtract the probability of the combined outcome that isn't part of that event from 1:

    straight P open parentheses A union B close parentheses equals 1 minus straight P open parentheses A to the power of italic apostrophe union B to the power of apostrophe close parentheses