3.14 M: Probability

• Match the probability value with its description:

  0 ↔️ Impossible

  Between 0 and 1 ↔️ Chance of occurring

  1 ↔️ Certain
• What is the formula for the probability of two independent events A and B occurring together?

  P(A \cap B) = P(A) \times P(B)</latex>
• What is the formula for the probability of two mutually exclusive events A and B occurring together?

  $P(A \cap B) =$$0$
• An event is a subset of the sample space.


  True
• What is a sample space in probability theory?

  All possible outcomes
• When rolling a 6-sided die, the sample space is {1, 2, 3, 4, 5, 6}

  True
• Order the following concepts in probability theory from most general to most specific:

  1️⃣ Sample Space

  2️⃣ Event
• What is the highest value a probability can have?

  1
• Two events are independent if the occurrence of one does not affect the likelihood of the other
• Mutually exclusive events cannot occur together.


  True
• What is the formula for the conditional probability of event A given event B has occurred?

  P(A|B) = \frac{P(A \cap B)}{P(B)}</latex>
• Probability is expressed as a number between 0 and 1.
• What is the definition of a sample space?

  All possible outcomes
• The sample space represents the universal set of all possible outcomes.
• What is the range of probability values?

  0 to 1
• What is the formula for the probability range of an event A?

  $0 ≤ P(A) ≤ 1$
• What is the formula for the complement probability of an event A?

  $P(A') =$$1 - P(A)$
• What is the formula for the intersection of two independent events A and B?

  $P(A \cap B) =$$P(A) \times P(B)$
• What is an example of mutually exclusive events?

  Rolling a 1 and a 6
• Flipping a coin twice is an example of independent events because the outcome of the first flip does not affect the second.
• The probability of the intersection of two mutually exclusive events is always zero.
• Match the example with the corresponding concept:

  Sample space ↔️ {1, 2, 3, 4, 5, 6}

  Event ↔️ {2, 4, 6}
• The probability of the complement of an event A is calculated as $1 - P(A)$.
• The conditional probability of drawing a king given that the card is a spade is $\frac{1}{13}$.
• Independent events are events where the outcome of one event does not depend on the outcome of the other event.
• An event with a probability of 0 is impossible to occur.

  True
• The probability of the entire sample space Ω is 1.

  True
• The probability of an event is the proportion of the sample space it occupies.


  True
• For dependent events, the intersection formula includes the conditional probability \(P(B|A)\), which means the probability of B given that A has occurred.
• Mutually exclusive events have a probability of 0 for their intersection.
• Rolling a 1 and a 6 on a single die are mutually exclusive events.


  True
• What does the sample space represent in probability theory?

  All possible outcomes
• What is the range of possible values for the probability of any event A?

  $0 \leq P(A) \leq 1$
• Probability values range from 0
• Probability expresses the uncertainty associated with random events.

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

  $P(A \cap B) =$$P(A) \times P(B|A)$
• The conditional probability of event A given event B has already occurred is written as $P(A|B)$
• Steps to calculate conditional probability:

  1️⃣ Calculate the joint probability $P(A \cap B)$

  2️⃣ Calculate the marginal probability $P(B)$

  3️⃣ Use the formula $P(A|B) =$$\frac{P(A \cap B)}{P(B)}$
• Match the event type with its definition:

  Mutually Exclusive ↔️ Cannot occur at the same time

  Independent ↔️ Outcome of one does not depend on the other

  Complementary ↔️ If one occurs, the other must
• The sample space is denoted as $\Omega$