Lesson 2: Probability Rules and Calculations
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Questions and Answers

Which approach is used to calculate the probabilities in the discussed course?

  • Subjective approach
  • Classical approach (correct)
  • Empirical approach
  • Bayesian approach

    • According to the law of large numbers, what can be expected as the number of trials increases?

  • The empirical approach will diverge from theoretical probabilities.
  • The relative frequencies will fluctuate wildly.
  • The empirical probability will approach the theoretical probability. (correct)
  • Outcomes will become less predictable.

    • What does it imply when the gambling establishment calculates a probability that is slightly higher than the implied probability?

  • They expect to lose money on bets.
  • They seek to ensure profit over time. (correct)
  • They aim to break even on bets.
  • They are uncertain about the outcome.

    • In the context of coin flips, what would be the expected outcome over 50 flips?

    <p>25 heads and 25 tails</p> Signup and view all the answers

    Which of the following factors can affect a team's likelihood of winning the Stanley Cup?

    <p>Player injuries</p> Signup and view all the answers

    What does a payout of $1000 for a $100 bet at +900 odds indicate about the betting market?

    <p>The team has a low probability of winning.</p> Signup and view all the answers

    What is the probability of rolling an even number given that the number rolled is greater than 2?

    <p>0.5</p> Signup and view all the answers

    Which of the following describes mutually exclusive events?

    <p>They cannot occur simultaneously.</p> Signup and view all the answers

    What does the formal addition rule for probabilities state?

    <p>P(A ∪ B) = P(A) + P(B) − P(A ∩ B)</p> Signup and view all the answers

    If the probability of event A is 0.4, event B is 0.5, and P(A ∩ B) is 0.1, what is P(A ∪ B)?

    <p>0.7</p> Signup and view all the answers

    What is the probability of rolling a number greater than 4 and less than 4 at the same time?

    <p>0</p> Signup and view all the answers

    Which formula applies when calculating the intersection of two events?

    <p>P(A ∩ B) = P(A) * P(B)</p> Signup and view all the answers

    In the example given, how many outcomes are there when rolling a number greater than 2?

    <p>4</p> Signup and view all the answers

    Using the formula for conditional probability, what is P(even # ∩ # > 2) if P(even # | # > 2) = 0.5?

    <p>2/6</p> Signup and view all the answers

    What does the formal multiplication rule for probabilities state about the intersection of two events A and B?

    <p>P(A ∩ B) = P(A) ∙ P(B|A)</p> Signup and view all the answers

    Why is it important to use P(B|A) instead of P(B) when calculating probabilities of dependent events?

    <p>Event A's occurrence may affect the likelihood of event B occurring</p> Signup and view all the answers

    In which scenario would the events be considered independent?

    <p>Rolling a die multiple times</p> Signup and view all the answers

    How can a contingency table help in calculating probabilities?

    <p>It simplifies the calculations by breaking down data into segments based on two variables</p> Signup and view all the answers

    What is the implication of having P(B|A) = P(B) for two events?

    <p>The events are independent of each other</p> Signup and view all the answers

    When can the multiplication rule be applied directly in probability calculations?

    <p>When events A and B are independent</p> Signup and view all the answers

    Which of the following best illustrates dependent events?

    <p>Choosing a person from a group and then selecting their favorite color</p> Signup and view all the answers

    What does the term 'intersection' refer to in probability?

    <p>The simultaneous occurrence of two events</p> Signup and view all the answers

    How does the probability obtained with replacement compare to that without replacement?

    <p>The difference appears at the 8th decimal place.</p> Signup and view all the answers

    In a probability tree for flipping a coin three times, how many possible outcomes are there?

    <p>8</p> Signup and view all the answers

    What kind of events does the order matter in a probability tree?

    <p>Dependent events only</p> Signup and view all the answers

    What is the probability of obtaining Heads or Tails in a single flip of a coin?

    <p>0.5</p> Signup and view all the answers

    If an event is dependent in a probability tree, how are future outcomes affected?

    <p>They could change both probabilities and possible events.</p> Signup and view all the answers

    When using a probability tree, what is the primary purpose of laying out events sequentially?

    <p>To simplify complex calculations.</p> Signup and view all the answers

    What type of sampling can be assumed when working with populations due to negligible differences at certain decimal points?

    <p>Sampling with replacement</p> Signup and view all the answers

    What is the probability of getting three heads (HHH) in three coin flips?

    <p>0.125</p> Signup and view all the answers

    Which outcome has the same probability as HHT in three coin flips?

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

    What is the total number of different outcomes when flipping a coin three times?

    <p>8</p> Signup and view all the answers

    If TTH represents tails on the first two flips and heads on the last, what is its probability?

    <p>0.125</p> Signup and view all the answers

    Which of these probabilities corresponds to getting all tails (TTT) in three flips?

    <p>0.125</p> Signup and view all the answers

    How would the probability of obtaining the sequence HTH be expressed?

    <p>$P(HTH) = 0.125$</p> Signup and view all the answers

    Which of the following probabilities is incorrect for the sequences from three coin flips?

    <p>P(HHT) = 0.25</p> Signup and view all the answers

    What does each terminal node represent in the context of three coin flips?

    <p>Intersection of outcomes along the path</p> Signup and view all the answers

    What is the probability of getting exactly one head and two tails in three flips?

    <p>0.375</p> Signup and view all the answers

    If a sequence has more tails than heads in three flips, what implication does this have for the probabilities?

    <p>Probability equal to sequences with more heads</p> Signup and view all the answers

    Study Notes

    Probability Rules

    • Probability is a measure of the likelihood of an event occurring.
    • Probabilities range from 0 (impossible) to 1 (certain).
    • Probabilities can also be expressed as percentages (multiply by 100).
    • Probability is the long-term relative frequency of an event.
    • Probability provides the expectation of the likelihood of an event.
    • Probabilities can be determined using several approaches: classical (theoretical), empirical (relative frequency), or subjective.

    Key Definitions

    • An experiment is a process that produces results.
    • An outcome is a particular result of an experiment.
    • A sample space is the set of all possible outcomes.
    • A simple event is an outcome.
    • A compound event is a combination of outcomes.
    • Event A (denoted as A) is one or more outcomes of the experiment.
    • An event is a collection of outcomes.

    Types of Probability

    • Classical (theoretical): Assumes all outcomes are equally likely.
      • Probability(A) = (number of favorable outcomes) / (total number of possible outcomes)
    • Empirical (relative frequency): Based on observed data.
      • Probability(A) = (number of times A occurred) / (total number of trials)
    • Subjective: Based on judgment and experience. Example: likelihood of events occurring.

    Probability Rules

    • Complements: The complement of an event is everything not in that event. P(A) + P(A') = 1.

    • Intersections: The intersection of two events A and B (denoted A∩B) is when both A and B happen. P(A ∩ B) = P(A) * P(B|A)

    • Unions: The union of events A and B (denoted A∪B) is when A, B, or both A and B happen. P(A∪B) = P(A) + P(B) - P(A∩B)

    • Mutually Exclusive: Events that cannot happen at the same time. If A and B are mutually exclusive, P(A∩B) = 0.

    • Conditional Probability: The probability of event B happening, given that event A has already happened: P(B|A) = P(A∩B) / P(A)

    • Multiplication Rule: The intersection of two events -- that is when event A and event B both need to occur,

      P(A and B) = P(A) x P(B given A)

    • Independent events: If the occurrence of one event does not affect the probability of another event. P(A ∩ B) = P(A) * P(B)

    • Dependent events: If the occurrence of one event affects the probability of another event.

    • Sampling with replacement: Choosing items from a set, returning them, and then repeating the process. Each event is independent.

    • Sampling without replacement: Choosing items from a set and not returning them. Each event is dependent.

    Probability Calculations

    • Contingency Tables: Tables used to summarize categorical data.
    • Probability Trees: Visual models that show the different outcomes of events that occur in sequence.

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    Description

    Apply probability rules to calculate probabilities of simple and compound events.

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