2.3 Probability

• Order the three perspectives on defining probability from most objective to most subjective.

  1️⃣ Frequentist Probability

  2️⃣ Classical Probability

  3️⃣ Subjective Probability
• What is the definition of a sample space?

  All possible outcomes
• Frequentist probability requires a large number of trials.


  True
• What does the union of two events represent in probability?

  A or B occurring
• What is the condition for two events to be considered independent?

  One does not affect the other
• What is the formula for the complement of an event A?

  $P(A') =$$1 - P(A)$
• Match the type of event with its formula:

  Independent Events ↔️ $P(A \cap B) =$$P(A) \times P(B)$

  Dependent Events ↔️ $P(A \cap B) =$$P(A) \times P(B|A)$
• Match the probability concept with its definition:

  Sample Space ↔️ The set of all possible outcomes

  Event ↔️ A subset of the sample space

  Probability ↔️ The likelihood of an event occurring
• What is the intersection of the events A and B in the example above, where A is rolling an even number and B is rolling a number less than 4 on a 6-sided die?

  $P(A \cap B) =$$\frac{1}{6}$
• Venn diagrams are used to visually represent set operations and calculate probabilities.

  True
• The complement of an event A, denoted as $A'$, represents the probability of A not occurring.


  True
• What is the formula for the intersection of two independent events A and B?

  $P(A \cap B) =$$P(A) \times P(B)$
• Match the type of event with its defining characteristic:

  Independent Events ↔️ Occurrence of one event does not affect the other

  Dependent Events ↔️ Occurrence of one event affects the other
• What is the key difference between mutually exclusive and non-mutually exclusive events?

  Non-mutually exclusive events can occur simultaneously
• Events are mutually exclusive if they cannot occur simultaneously
• Non-mutually exclusive events can occur simultaneously
• Match the type of event with its property:

  Mutually Exclusive ↔️ Events cannot occur simultaneously

  Non-Mutually Exclusive ↔️ Events can occur simultaneously
• What is the formula for Bayes' Theorem?

  P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}</latex>
• What is the formula for calculating the number of permutations of n distinct objects taken r at a time?

  $P(n, r) =$$\frac{n!}{(n - r)!}$
• Probability is a measure of the likelihood of an event occurring, given a set of possible outcomes
• What is the range of values for probability?

  0 to 1
• Classical probability assumes all outcomes in the sample space are equally likely.


  True
• An event is a subset of the sample
• What is the sample space when tossing a coin?

  {Heads, Tails}
• What is an example of a sample space in probability?

  {Heads, Tails}
• The intersection of two events represents the probability of both events occurring.


  True
• The probability of two independent events A and B occurring together is calculated as P(A \cap B) = P(A) \times P(B)</latex>


  True
• Independent events are those where the occurrence of one event affects the probability of the other.

  False
• Probability is expressed as a fraction between 0 and 1, where 0 means impossible and 1 means certain.


  True
• Venn diagrams can visually represent set operations to calculate probabilities.


  True
• What is the formula for the union of two events using set notation?

  $P(A ∪ B) =$$P(A) +$$P(B) - P(A ∩ B)$
• What does the union of two events $A ∪ B$ represent in probability?


  Probability of A or B
• If A is the event of rolling an even number on a 6-sided die, what is P(A)</latex>?

  $\frac{1}{2}$
• Dependent events occur when the outcome of one event influences the probability of the other event.

  True
• What is the probability of the intersection of two mutually exclusive events A and B?

  $P(A \cap B) =$$0$
• Example of independent and dependent events:

  1️⃣ Independent: Tossing a coin and rolling a die

  2️⃣ Dependent: Drawing a card from a deck without replacement
• The probability of two mutually exclusive events A and B occurring together is $P(A \cap B) =$$0$


  True
• The probability of two non-mutually exclusive events A and B occurring together is $P(A \cap B) \neq 0$


  True
• Conditional probability is denoted as $P(A|B)$, which means the probability of event A given event B.


  True
• Match the concept with its use:

  Conditional Probability ↔️ Calculate probability given an event

  Bayes' Theorem ↔️ Reverse the conditional probability