Probability Chapter

Probability

• Probability is used to quantify the likelihood or chance that an outcome of a random experiment will occur.
• It is the branch of mathematics concerning events and numerical descriptions of how likely they are to happen.
• Probability is essential in assessing risks and making better decisions throughout scientific and engineering disciplines.

Sample Spaces and Events

• A random experiment is an experiment that can result in different outcomes, even though it is repeated in the same manner every time.
• A permutation can be constructed by selecting the element to be placed in the first position of the sequence from the n elements, then selecting the element for the second position from the n-1 remaining elements, and so forth.
• Permutations are sometimes referred to as linear permutations.

Combinations

• A combination is the number of subsets of r elements that can be selected from a set of n elements, where order is not important.
• Every subset of r elements can be indicated by listing the elements in the set and marking each element with a “*” if it is to be included in the subset.

Interpretations and Axioms of Probability

• The first axiom states that the probability of the sample space is equal to 1.
• The second axiom states that a probability is nonnegative.
• The third axiom states that for every collection of mutually exclusive events, the probability of their union is the sum of the individual probabilities.

Total Probability Rule

• The total probability rule is used to find the total probability of either of two events occurring.
• P(A) + P(B) - P(A∩B) = P(A) + P(B) - P(A)P(B) if they are not mutually exclusive.

Multiple Events

• To find the total probability of any of the events A1, A2, …, An occurring, we apply the principle of inclusion-exclusion, summing individual probabilities and subtracting the probabilities of all possible intersections.
• P(F) = P(F|H)P(H) + P(F|M)P(M) + P(F|L)P(L)

Independence of Conditional Probabilities

• Two events A and B are independent if P(A|B) = P(A) and P(B|A) = P(B).
• If P(A|B) = P(A) and P(B|A) = P(B), then A and B are independent.

Bayes' Theorem

• Bayes' theorem is used to find the probability of another conditional event given the initial conditional event.
• The theorem was developed by Thomas Bayes around the 1700s.

Probability is a branch of mathematics that quantifies the likelihood of an outcome in a random experiment. It helps assess risks and make better decisions in scientific and engineering disciplines.