5.2 Combined Events

• How is probability calculated?

  Favorable outcomes / total outcomes
• The probability of getting heads in a coin toss is 1/2.
• The probability of rolling a 4 on a standard dice is 1/6
• What does probability measure?

  Likelihood of an event
• The probability of getting heads in a coin toss is 50%
• Independent events are influenced by each other.

  False
• What is the key difference between independent and dependent events?

  Dependence of outcomes
• The outcomes of independent events do not affect each other
• Match the event type with an example:

  Independent event ↔️ Flipping a coin

  Dependent event ↔️ Drawing cards without replacement
• What is the formula for calculating the probability of independent events using AND?

  P(A \text{ and } B) = P(A) \times P(B)</latex>
• The conditional probability of event B given event A is written as $P(B|A)$
• For dependent events, the conditional probability must be used to calculate combined probabilities.
• Match the event type with its probability calculation:

  Independent event ↔️ $P(A \text{ and } B) =$$P(A) \times P(B)$

  Dependent event ↔️ $P(A \text{ and } B) =$$P(A) \times P(B|A)$
• For independent events, the probability of both events occurring is the product
• For dependent events, the probability of both events occurring is P(A) \times P(B|A)</latex>, where $P(B|A)$ is the conditional probability of event B occurring given that event A has occurred.
• The key difference between independent and dependent events is that independent events use multiplication, while dependent events use conditional probability.
• In independent events, outcomes do not affect each other.
• If the probability of getting a head on a coin toss is 1/2, and the probability of rolling a 4 on a dice is 1/6, the probability of both events occurring is 1/12.
• Steps to calculate the probability of combined events using OR for independent events

  1️⃣ Add the individual probabilities of event A and event B

  2️⃣ Subtract the probability of both A and B occurring
• For dependent events, the probability of either event A or event B occurring is calculated using the union rule.
• In dependent events, the conditional probability P(B|A) must be used in calculations.
• If the probability of getting a head on a coin toss is 1/2, and the probability of rolling a 4 on a dice is 1/6, the probability of getting a head OR a 4 is 2/3.
• Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
• Match the event with its probability:

  Coin toss (Heads) ↔️ 1/2 or50%

  Dice roll (Getting a 4) ↔️ 1/6 or ~16.67%
• Independent events are events where the outcome of one event does not affect the probability of another event occurring, meaning the events are separate.
• In dependent events, the probability of the second event depends on the outcome of the first event.
• Independent events are events where the outcome of one event does not affect the probability
• Dependent events are events where the outcome of one event affects the probability of another event occurring.
• Match the event type with its description:

  Independent Events ↔️ Outcomes do not affect each other

  Dependent Events ↔️ Outcomes are related
• Flipping a coin and then rolling a dice is an example of independent events.
• Drawing a card from a deck without replacement is an example of dependent events.
• Steps to calculate the probability of combined events using AND for dependent events:

  1️⃣ Calculate P(A)

  2️⃣ Calculate P(B|A)

  3️⃣ $P(A \text{ and } B) =$$P(A) \times P(B|A)$
• The probability of getting heads on a coin and a 4 on a dice is 1/12.
• The probability of drawing two aces from a deck without replacement is 1/221.
• In the formula for dependent events using AND, P(B|A)</latex> is the conditional probability of event B occurring given that event A has occurred.
• Match the event type with its AND formula:

  Independent Events ↔️ $P(A \text{ and } B) =$$P(A) \times P(B)$

  Dependent Events ↔️ $P(A \text{ and } B) =$$P(A) \times P(B|A)$
• For independent events using OR, the probability of either A or B occurring is given by P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)</latex>, where $P(A \text{ and } B) =$$P(A) \times P(B)$.product
• The probability of one event in independent events is influenced by the outcome of another event.

  False
• If the probability of getting a head on a coin toss is 1/2 and rolling a 4 on a dice is 1/6, the probability of both events occurring is 1/12
• The calculation of combined probabilities using OR requires distinguishing between independent and dependent events.