probability important theorms

 Certainly, here are explanations and solutions for some important probability theorems:

1. Addition Rule for Disjoint (Mutually Exclusive) Events: 

  Theorem: If A and B are disjoint (mutually exclusive) events, then the probability of either event occurring is the sum of their individual probabilities.

Mathematically:  P (A or B) = P(A) + P(B)

  Solution: Let's say we roll a fair six-sided die. What is the probability of rolling a 2 or a 4?

 Solution: P (rolling a 2) = 1/6, P (rolling a 4) = 1/6

 P (rolling a 2 or a 4) = P (rolling a 2) + P (rolling a 4) = 1/6 + 1/6 = 1/3

2. Multiplication Rule for Independent Events

Theorem:  If A and B are independent events, then the probability of both events occurring is the product of their individual probabilities.

Mathematically:  P (A and B) = P(A) * P(B)

Solution: Consider drawing a card from a standard deck (without replacement). What is the probability of drawing a spade and then drawing a heart?

Solution: P (drawing a spade) = 13/52 (first draw)

   P (drawing a heart) = 13/51 (second draw, since one card has been removed)

   P (drawing a spade and then a heart) = P (drawing a spade) * P (drawing a heart) = (13/52) * (13/51) ≈ 0.064

3. Complementary Probability Rule:

Theorem: The probability of an event not occurring is 1 minus the probability of the event occurring.

Mathematically: P(A') = 1 - P(A)

Solution: What is the probability of rolling a number less than 5 on a fair six-sided die?

Solution: P (rolling a number less than 5) = 1 - P (rolling a 5 or 6) = 1 - (1/6 + 1/6) = 2/3

4. Conditional Probability:

Theorem:  The probability of event A occurring given that event B has already occurred is the probability of both events A and B occurring divided by the probability of event B occurring.

Mathematically: P(A|B) = P (A and B) / P(B)

Solution: In a bag, there are 5 red marbles and 3 blue marbles. If one marble is drawn at random and is found to be red, what is the probability that it's also blue?

Solution: P (red and blue) = 0 (since a marble cannot be both red and blue)

   P(blue) = 3 / 8 (total blue marbles)

   P(red|blue) = P (red and blue) / P(blue) = 0 / (3 / 8) = 0

5. Law of Total Probability:

Theorem: If B1, B2, ..., Bn are mutually exclusive and exhaustive events, then the probability of event A occurring is the sum of the probabilities of A occurring given each Bi, multiplied by the probability of Bi.

Mathematically: P(A) = Σ [P(A|Bi) * P(Bi)]

Solution: An urn contains 3 red balls and 5 blue balls. A second urn contains 4 red balls and 6 blue balls. If one urn is selected at random and a ball is drawn from it, what is the probability of drawing a blue ball?

Solution: Let B1 be selecting the first urn, and B2 be selecting the second urn.

   P(B1) = 1/2 (assuming equal probability of selecting either urn)

   P(B2) = 1/2

   P(blue|B1) = 5 / (3 + 5) = 5/8

   P(blue|B2) = 6 / (4 + 6) = 6/10 = 3/5

   P(blue) = P(blue|B1) * P(B1) + P(blue|B2) * P(B2) = (5/8) * (1/2) + (3/5) * (1/2) = 19/40

These theorems provide a foundation for understanding probability and solving various types of probability problems. Feel free to practice with different scenarios and problems to strengthen your understanding of these concepts.