2.3.4 Independent events

• What are independent events in probability theory?

  Events where one does not affect the other
• Two events are independent if the occurrence of one changes the probability of the other happening.

  False
• If events A</latex> and $B$ are independent, then $P(A \cap B) =$$P(A) \times P(B)$ is the formula to calculate the probability of both events occurring together
• What does $P(A)$ represent in the formula for independent events?


  Probability of event A
• What does $P(A \cap B)$ represent in the formula for independent events?


  Probability of both A and B
• The probability of getting heads on both flips of a fair coin is 0.25.
• Mathematically, independent events A</latex> and $B$ satisfy the condition $P(A \cap B) =$$P(A) \times P(B)$, which states that the probability of both events occurring is the product of their individual probabilities
• The event of getting heads on the first flip of a coin is independent of getting heads on the second flip.
• Steps to calculate the probability of two independent events occurring together

  1️⃣ Identify the two independent events

  2️⃣ Determine the probability of each event

  3️⃣ Multiply the individual probabilities

  4️⃣ Calculate the product to find $P(A \cap B)$
• If $P(A) =$$0.6$ and $P(B) =$$0.4$, what is $P(A \cap B)$ for independent events?


  0.24
• Two events are independent if the probability of both occurring is equal to the product of their individual probabilities.
• Suppose $P(A) =$$0.4$, P(B) = 0.5</latex>, and $P(A \cap B) =$$0.2$. Are $A$ and $B$ independent?


  Yes
• To solve problems involving independent events, we use the formula $P(A \cap B) =$$P(A) \times P(B)$, where $P(A)$ and $P(B)$ are the probabilities of events $A$ and $B$, and $P(A \cap B)$ is the probability of both together
• Two independent events, $A$ and $B$, have probabilities $P(A) =$$0.6$ and $P(B) =$$0.4$. What is $P(A \cap B)$?


  0.24
• Match the event with its probability:

  $A$ ↔️ $0.6$

  $B$ ↔️ $0.4$

  $A \cap B$ ↔️ $0.24$