11 Questions
What is the condition for two events A and B to be independent?
P(A|B) = P(A) and P(B|A) = P(B)
What is the purpose of Bayes' Theorem?
To find the probability of another conditional event given the total sum of partitions' probability
If P(A) = 0.4, P(B) = 0.8, and P(A B) = 0.36, are events A and B independent?
No, because P(A B) ≠ P(A)P(B)
What is the probability of event A in Example 1?
1/3
What is the probability of event C in Example 1?
1/2
If P(A) = 0.4, P(B) = 0.8, and P(A|B) = 0.45, are events A and B independent?
No, because P(A|B) ≠ P(A)
If P(A) = 0.4, P(B) = 0.8, and P(A B) = 0.32, are events A and B independent?
Yes, because P(A B) = P(A)P(B)
Who developed Bayes' Theorem?
Thomas Bayes
What is the formula to check for independence between events A and B?
P(A|B) = P(A) and P(B|A) = P(B)
What is the purpose of the denominator in Bayes' Theorem?
To find the total sum of partitions' probability
In Example 1, are events A and C independent?
Yes, because P(A|C) = P(A)
Study Notes
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.
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