2.3 Probability

• Bayes' Theorem is used to update beliefs when new evidence is presented.


  True
• The key difference between conditional probability and Bayes' Theorem is that Bayes' Theorem considers the probability of B|A
• What is the purpose of a tree diagram in probability calculations?

  Visualize and calculate probabilities
• For independent events A and B, the probability of both occurring is calculated as P(A and B) = P(A) × P(B).
• Probabilities close to 0 indicate less likely events, while probabilities close to 1 indicate very likely events.

  True
• What is the probability of rolling a 3 on a fair six-sided die?

  1/6
• What is the formula for the probability of a simple event?

  P(A) = Favorable outcomes / Total outcomes
• What is the formula for the intersection of two independent events A and B?

  P(A and B) = P(A) * P(B)
• A simple event consists of a single outcome.

  True
• Mutually exclusive events cannot occur at the same time.

  True
• P(B|A) represents the conditional probability of B given A.


  True
• Given P(A) = 0.4, what is P(A')?

  0.6
• Conditional probability is denoted as P(A|B), which is read as "the probability of A given B
• Bayes' Theorem is given by the formula P(A|B) = (P(B|A) * P(A)) / P(B)
• The probability of mutually exclusive events A or B is given by P(A or B) = P(A) + P(B)
• In dependent events, the outcome of one event affects the outcome of the other.


  True
• For dependent events, the probability of A and B is calculated as P(A and B) = P(A) * P(B|A)
• Match the type of event with its formula:

  Simple Event ↔️ P(A) = Favorable outcomes / Total outcomes

  Union (Mutually Exclusive) ↔️ P(A or B) = P(A) + P(B)

  Union (Non-Mutually Exclusive) ↔️ P(A or B) = P(A) + P(B) - P(A and B)

  Intersection (Independent) ↔️ P(A and B) = P(A) * P(B)

  Intersection (Dependent) ↔️ P(A and B) = P(A) * P(B|A)

  Complement ↔️ P(A') = 1 - P(A)
• For mutually exclusive events, P(A or B) is calculated as P(A) + P(B).


  True
• The probability of the complement of A, P(A'), is equal to 1 - P(A)
• The formula for conditional probability is P(A|B) = P(A and B) / P(B)
• Steps to use a tree diagram

  1️⃣ Identify the events and their probabilities

  2️⃣ Draw branches for each possible outcome

  3️⃣ Multiply the probabilities along each branch

  4️⃣ Add up the probabilities of the individual outcomes
• Match the type of probability distribution with its characteristic:

  Discrete ↔️ Finite or countable outcomes

  Continuous ↔️ Infinite outcomes over a range
• Discrete probability distributions describe the probability of a finite or countable number of outcomes
• What type of function is used to calculate probabilities in a continuous probability distribution?

  Probability density function (PDF)
• How are probabilities calculated in a discrete probability distribution?

  Sum of individual outcome probabilities
• The key difference between discrete and continuous distributions is the nature of their possible outcomes.


  True
• The expected value of a continuous distribution is calculated using the integral of x multiplied by the PDF.

  True
• The variance of a probability distribution measures how spread out the possible values are from the mean.
• The standard deviation is the square root of the variance and measures the typical deviation from the mean.


  True
• What is the numerical range for probability values?

  0 to 1
• The sample space is the set of all possible outcomes of an experiment.

  True
• What is the formula for the probability of dependent events A and B?

  P(A) × P(B|A)
• The probability of a simple event is calculated as P(A) = Number of favorable outcomes / Total number of possible outcomes.
• For independent events A and B, the probability of A and B is P(A and B) = P(A) * P(B).


  True
• Match the event type with its formula and condition:

  Simple Event ↔️ P(A) = Favorable outcomes / Total outcomes ||| -

  Union (Mutually Exclusive) ↔️ P(A or B) = P(A) + P(B) ||| Mutually Exclusive

  Intersection (Independent) ↔️ P(A and B) = P(A) * P(B) ||| Independent
• What is the formula for the intersection of two dependent events A and B?

  P(A and B) = P(A) * P(B|A)
• How is the probability of a simple event calculated?

  Favorable outcomes / Total outcomes
• What is the formula for the union of two mutually exclusive events A and B?

  P(A or B) = P(A) + P(B)
• What is the complement of an event A?

  Outcomes not in A