2.3.3 Conditional probability

• What is conditional probability defined as?

  Probability of A given B
• Conditional probability is denoted as P(A|B)
• The conditional probability formula is $P(A|B) =$$\frac{P(A \cap B)}{P(B)}$
• What does $P(A \cap B)$ represent in the conditional probability formula?


  Both A and B occur
• The conditional probability formula is valid only if $P(B)$ is not equal to zero
• What is the first step to solve a conditional probability problem using the formula?

  Identify events A and B
• Steps to solve conditional probability problems using the formula

  1️⃣ Identify events A and B

  2️⃣ Find P(A \cap B)</latex>, the probability of both A and B occurring

  3️⃣ Find $P(B)$, the probability of event B

  4️⃣ Apply the formula to calculate $P(A|B)$
• In the example of a survey where 20% read newspapers, 40% watch TV, and 10% do both, what is $P(A \cap B)$?


  0.1
• What does $P(A \cap B)$ represent in the conditional probability context?


  Both A and B occur
• In a bag containing 3 red and 2 blue balls, if a red ball is drawn and not replaced, the event of drawing another red ball is called conditional
• What does the event A represent in the example about newspapers and TV?

  Watches TV
• The event B represents reading newspapers
• $P(A \cap B)$ is the probability that both A and B occur.
• What is the value of $P(B)$ in the example about newspapers and TV?


  0.2
• The conditional probability P(A|B)</latex> is calculated as $\frac{0.1}{0.2}$, which equals 0.5
• Steps to solve a conditional probability problem

  1️⃣ Identify events A and B

  2️⃣ Find $P(A \cap B)$, the probability of both A and B occurring

  3️⃣ Find $P(B)$, the probability of event B occurring

  4️⃣ Apply the formula $P(A|B) =$$\frac{P(A \cap B)}{P(B)}$
• What is the definition of conditional probability?

  Probability of A given B
• $P(A|B)$ is calculated by dividing $P(A \cap B)$ by $P(B)$
• What does $P(A \cap B)$ represent in the conditional probability formula?


  Probability of both A and B
• P(B)</latex> is the probability of event B occurring.
• In the bag example, what is the probability of drawing a red ball first?

  $\frac{3}{5}$
• $P(A \cap B)$ in the bag example is calculated as \frac{3}{5} \times \frac{2}{4}</latex>
• What is the value of $P(A|B)$ in the bag example?


  $\frac{1}{2}$
• The conditional probability formula requires that $P(B)$ is not equal to 0
• $P(B)$ is the probability of event B occurring.
• In the second bag example, what is the probability of drawing a red ball first?

  $\frac{4}{10}$
• What is the first step in solving a conditional probability problem?

  Identify events A and B
• In the survey example, $P(A \cap B)$ is given as 0.1
• The conditional probability $P(A|B)$ in the survey example is 0.5.
• What is the formula for conditional probability $P(A|B)$?


  $P(A|B) =$$\frac{P(A \cap B)}{P(B)}$
• The conditional probability formula remains the same regardless of the order of events.
• The conditional probability formula is P(A|B) = \frac{P(A \cap B)}{P(B)}</latex>
• What two probabilities must be calculated before applying the conditional probability formula?

  $P(A \cap B)$ and $P(B)$
• Steps to apply conditional probability to real-world scenarios

  1️⃣ Identify events A and B

  2️⃣ Calculate $P(A \cap B)$

  3️⃣ Calculate $P(B)$

  4️⃣ Apply the conditional probability formula
• Why is it important to distinguish between $P(A|B)$ and $P(B|A)$?


  The order of events matters
• When calculating conditional probability, dividing by $P(A \cap B)$ is correct.


  False
• What range should all probabilities fall within?

  0 to 1
• A probability can be negative.

  False