Probability Basics: Rolling Dice and Drawing Cards
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Questions and Answers

What is the sample space when rolling a six-sided die?

𝜉 = {1, 2, 3, 4, 5, 6}

The probability of an outcome is given by the formula: $Pr(outcome) = \frac{1}{______}$.

number of outcomes

What is the probability of rolling an odd number on a six-sided die?

  • 1/3
  • 1/4
  • 1/2 (correct)
  • 1/6

    • If a bag contains four purple balls and two green balls, what is the probability of drawing a green ball?

    <p>1/3</p> Signup and view all the answers

    The probability of tossing Heads is the same as the probability of tossing Tails if the coin is fair.

    <p>True</p> Signup and view all the answers

    What is a chance experiment?

    <p>A chance experiment is a process that can be repeated many times.</p> Signup and view all the answers

    What are the possible outcomes when tossing a coin?

    <p>Heads or Tails</p> Signup and view all the answers

    Which event describes rolling a number greater than 5 on a six-sided die?

    <p>C = {6}</p> Signup and view all the answers

    The favourable outcomes for rolling an even number on a die are {blank}.

    <p>{2, 4, 6}</p> Signup and view all the answers

    What is the probability of drawing a king from a standard deck of cards?

    <p>1/13</p> Signup and view all the answers

    How is the relative frequency calculated?

    <p>By dividing the frequency of the outcome by the total number of trials.</p> Signup and view all the answers

    What is the formula for experimental probability?

    <p>Experimental probability = frequency of an outcome / total number of trials</p> Signup and view all the answers

    If a spinner is spun 100 times and results in frequencies of 7, 26, 9, and 58, what is the experimental probability for outcome 1?

    <p>0.07</p> Signup and view all the answers

    What is the expected number of results for an event that happens with a probability of 0.5 after 100 trials?

    <p>50</p> Signup and view all the answers

    In Tattslotto, if six numbers are drawn from the numbers 1 to 45, what is the probability of winning by buying one ticket?

    <p>1/8,145,060</p> Signup and view all the answers

    If a coin is flipped 400 times, what is the expected number of Tails?

    <p>200</p> Signup and view all the answers

    If the probability of Tory winning a match is 0.6 and she plays 10 matches, how many matches is she expected to win?

    <p>6</p> Signup and view all the answers

    What is the experimental probability of rolling a one on a die if it is rolled 120 times and the outcome is 20?

    <p>1/6</p> Signup and view all the answers

    The expected number of outcomes is calculated from the formula: Expected number = probability of outcome × number of trials.

    <p>True</p> Signup and view all the answers

    In an experiment, if the relative frequency of Heads is 0.8 after tossing a coin five times, this is a reasonable probability.

    <p>False</p> Signup and view all the answers

    How do you determine the angle size of each section of a spinner based on the frequency results?

    <p>By calculating the proportion of each section's frequency to the total frequency and then multiplying by 360 degrees.</p> Signup and view all the answers

    What is the complement of the event 'rolling an odd number' when rolling a die?

    <p>{2, 4, 6}</p> Signup and view all the answers

    What is the complement of A = {evens} for the sample space $\xi = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$?

    <p>{1, 3, 5, 7, 9}</p> Signup and view all the answers

    A six-sided die is rolled, and the number uppermost is noted. The event of rolling an even number is:

    <p>{2, 4, 6}</p> Signup and view all the answers

    How many trials were there if a coin is tossed with Heads appearing 72 times and Tails 28 times?

    Signup and view all the answers

    What is the experimental probability for tossing a Tail?

    <p>This is calculated as the number of Tails divided by the total number of trials.</p> Signup and view all the answers

    What is the theoretical probability for tossing a Tail with a fair coin?

    Signup and view all the answers

    What is the difference in this experiment between the theoretical and experimental probabilities for tossing a Tail?

    <p>You find the difference by subtracting the theoretical probability from the experimental probability.</p> Signup and view all the answers

    Calculate the exact probability of selecting a number that is a multiple of 3 from the set {1, 2, ..., 30}.

    Signup and view all the answers

    If a large number of trials is conducted in an experiment, the relative (experimental) frequency of each outcome will be very close to its theoretical probability.

    <p>True</p> Signup and view all the answers

    What is the probability that two aces are chosen when drawing two cards from a shuffled deck with replacement?

    Signup and view all the answers

    If ξ = {numbers between 1 and 20 inclusive}, which set B is equal to {multiples of 3 numbers between 1 and 20}?

    <p>{5, 19}</p> Signup and view all the answers

    Study Notes

    The Language of Probability

    • Probability measures the likelihood of an event occurring.
    • An event with a probability of 1 is certain to occur.
    • An event with a probability of 0 is impossible.
    • Events with probabilities between 0 and 1 are possible.
    • Events with probabilities between 0.5 and 1 are likely to occur.
    • Events with probabilities between 0 and 0.5 are unlikely to occur.

    Key Terms of Probability

    • A chance experiment is a process that can be repeated multiple times.
    • A trial is one performance of a chance experiment resulting in an outcome.
    • An outcome is the result of a single trial.
    • The sample space (𝜉) is the set of all possible outcomes.
    • An event is a favourable outcome within the sample space, often represented by a capital letter.

    Favourable Outcomes

    • A favourable outcome is any outcome that is part of the event
    • Example: For a dice roll, rolling an even number has favourable outcomes of 2, 4, and 6

    Sample Space

    • Sample Space (𝜉) represents all possible outcomes of a chance experiment
    • Example: for rolling a die, the Sample Space is {1, 2, 3, 4, 5, 6}

    Events

    • Events are a subset of the sample space, describing specific outcomes
    • Example: rolling a 4 on a die is an event represented as {4}
    • Example: rolling at least 5 on a die is the event {5, 6}

    Theoretical Probability

    • Represents the likelihood of an event occurring
    • Calculated by dividing the number of favorable outcomes by the total number of outcomes
    • Formula: Pr(event A) = (number of favorable outcomes)/(total number of outcomes)

    Relative Frequency

    • The ratio of the frequency of an outcome in a chance experiment to the total number of trials
    • Calculated by dividing the frequency of an outcome by the total number of trials
    • Formula: Relative frequency = (frequency of an outcome)/(total number of trials)

    Experimental Probability

    • The relative frequency of an outcome based on repeated trials
    • Used when theoretical probability cannot be determined
    • Formula: Experimental probability = (frequency of an outcome)/(total number of trials)

    Expected number of results

    • The expected number of favorable outcomes can be calculated using the formula: Expected number = probability of outcome × number of trials
    • The probability can be either theoretical or experimental.

    Determining the Complement

    • The complement of an event, A', contains all outcomes that are not in event A.
    • The following equation holds true: Pr(A) + Pr(A') = 1

    Venn diagrams: The intersection of events

    • Venn diagrams use rectangles and circles to represent sets of objects or numbers
    • The entire sample space is represented by the rectangle symbol 𝜉
    • Circles within the rectangle contain all possible outcomes for a given event
    • Overlapping regions between circles signify outcomes that belong to multiple events.

    Intersection of Events

    • The intersection of two events is represented by the overlapping region of their Venn diagrams, symbolized by the '∩' symbol
    • It captures outcomes that are present in both events
    • For example, A ∩ B represents the outcomes that are in event A and in event B

    Venn Diagram Regions Explained

    • A ∩ B′: Outcomes found in event A but not in event B
    • A ∩ B: Outcomes found in both event A and event B
    • A′ ∩ B: Outcomes found in event B but not in event A
    • A′ ∩ B′: Outcomes not found in either event A or event B

    Two-Way Tables

    • Two-way tables are alternative representations of information presented in Venn diagrams
    • They offer a structured way to organize data by categorizing outcomes based on two different attributes
    • Each cell within a two-way table represents the intersection of specific categories
    • The total number of outcomes in each row and column is calculated and displayed
    • The relationship between Venn diagrams and two-way tables is demonstrated, showing how they complement each other in presenting data

    Number of Outcomes

    • The number of outcomes within a specific event is represented by the notation "n(event)". For example, 'n(A)' represents the number of outcomes in event A.
    • The notation can be used to determine the number of outcomes found in intersections, such as 'n(A ∩ B)' indicating the number of outcomes present in both event A and event B

    Calculating Union of Events

    • The union of two events, represented by 'A ∪ B', includes all outcomes present in either event A or event B
    • The union captures all outcomes that are within the combined regions of events A and B on a Venn diagram.

    Probability

    • Probability is the chance of an event happening.
    • It is calculated by considering the favorable outcomes (those representing the event of interest) and dividing them by the total number of outcomes in the sample space.
    • For example, Pr(A) represents the probability of event A, calculated as n(A)/n(𝜉), with 𝜉 representing the sample space.

    Venn Diagrams

    • Venn diagrams are used to visually represent sets and their relationships.
    • Each set is represented by a circle.
    • The overlapping region of circles shows the elements that are common to both sets.
    • Elements outside the circles represent those not belonging to either set.

    Set Notation

    • A ∩ B represents the intersection of sets A and B, containing elements common to both.
    • A ∪ B represents the union of sets A and B, containing all elements from both sets.
    • A ∩ B′ denotes the elements in set A but not in set B.
    • A′ ∪ B′ represents elements not in set A or set B, equivalent to the complement of the intersection of A and B.

    Probability with Venn Diagrams

    • To calculate the probability of an event, divide the number of favorable outcomes by the total number of outcomes in a sample space.
    • Venn diagrams can be used to visualize events and their probabilities.
    • The areas within the circles represent the probabilities of individual events.
    • The overlapping area represents the probability of the intersection of events.

    Example using Basketball and Tennis

    • A class of 24 students includes 11 playing basketball (B), 7 playing tennis (T), and 4 playing both.
    • This information can be represented in a Venn diagram.
    • n(B ∩ T) = 4 indicates the number of students playing both sports.
    • n(B ∩ T′) = 7 represents students playing basketball but not tennis.
    • n(T ∩ B′) = 3 represents students playing tennis but not basketball.
    • n(B′ ∩ T′) = 10 represents students playing neither.
    • This information can be used to calculate probabilities like Pr(B) = 11/24, representing the probability of a student playing basketball.
    • Similarly, Pr(T ∪ B) = 14/24 represents the probability of a student playing tennis or basketball.
    • The probability of a student playing tennis or basketball but not both is found by adding n(B ∩ T′) and n(B′ ∩ T) and dividing by the total number of students, resulting in 10/24.

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    Probability PDF

    Description

    Test your understanding of probability with this quiz focused on rolling dice and drawing cards. You'll explore concepts such as sample space, event outcomes, and computation of probabilities for various scenarios. Perfect for students looking to solidify their grasp on probability fundamentals.

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