Probability

Probability is the branch of mathematics that quantifies uncertainty and measures the likelihood of events. It ranges from 0 (impossible) to 1 (certain). For discrete sample spaces, probability is defined as the ratio of favorable outcomes to total possible outcomes. Fundamental rules include the addition rule for unions of events, the multiplication rule for intersections, and conditional probability for dependent events. Bayes’ theorem provides a way to update probabilities based on new information. Engineers and scientists use probability to model reliability, risk, and uncertainty in systems such as quality control, communication networks, and structural safety.

Governing FormulaBasic probability: P(A) = n(A)/n(S) Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Multiplication rule: P(A ∩ B) = P(A) × P(B|A) Conditional probability: P(A|B) = P(A ∩ B)/P(B) Bayes’ theorem: P(A|B) = [P(B|A) × P(A)] / P(B)

Knowledge Check

10 Questions

1.A fair six-sided die is rolled once. What is the probability of rolling an even number?

2.Two coins are tossed simultaneously. What is the probability of getting exactly one head?

3.A card is drawn from a standard deck of 52 cards. What is the probability that it is a heart?

4.If P(A) = 0.4 and P(B) = 0.3, and A and B are mutually exclusive, what is P(A ∪ B)?

5.A bag contains 3 red balls and 5 blue balls. Two balls are drawn without replacement. What is the probability that both are red?

6.What is the probability of rolling a sum of 7 with two dice?

7.If P(A|B) = 0.5 and P(B) = 0.4, what is P(A ∩ B)?

8.Bayes’ theorem is used to calculate:

9.Two events are independent if:

10.A box contains 6 red, 4 blue, and 2 green marbles. One marble is drawn at random. What is the probability that it is not blue?