9516913
Description
Flashcards by hinal vaghela, updated more than 1 year ago
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Created by hinal vaghela
over 7 years ago
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| Question | Answer |
| Element | Is a possible outcome of an experiment |
| Event | Is a collection of one or more outcomes |
| Set | Is a collection of zero or more elements |
| Sample Space | Is the collection of all possible outcomes |
| Empty Space | Is a set containing zero element |
| Subset | IFF all elements in set D are also in set E |
| Union | Is the set of all elements that are contained in at least one of the sets |
| Intersection | Is the set of elements that are common for each set |
| Difference | Is the the set containing all elements that are in set F but not in set G |
| Mutually Exclusive | Is when two events do not occur at the same time |
| Disjoint | Same as mutually exclusive (when two events do not occur at the same time) |
| Complement | Is the set containing all of the elements in the sample space except the elements that are in T |
| Pr(anything) | Probabilities of anything are only defined between 0 and 1, inclusive |
| Pr(empty set) | Probability of the empty set is always 0 |
| Pr(sample space) | Probability of the sample space is always 1 |
| If A1, A2,... are disjoint events then the Pr(Union) is | Equal to the sum of the disjoint events |
| A c B, then | The Pr(A) =< Pr(B) Note: for A to be a subset of B, A has to contain an equal or fewer number of elements than B |
| The Additive Law of Probability |
Pr(A U B) = Pr(A) + Pr(B) - Pr(A & B)
If A and B are mutually exclusive, then Pr(A & B) = 0. Thus, Pr(A U B)= Pr(A) + Pr(B)
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| The union of the disjoint events must equal the sample space | Pr(A U A') = Pr(S) = 1 So, the Pr(A') = 1 - Pr(A) Note: Pr(A') is A complement. |
| The Law of Total Probability |
Consider n events, A1, A2,..., An which are disjoint. Pr(A1) + Pr(A2) + ... + Pr(An) = 1. Then A1, A2,..., An is a partition of the sample space, S.
Suppose event B exists in S. Then, the Pr(B) = Pr(B & A1) + Pr(B & A2) +...+ Pr(B & An).
Because B overlays the sets in the partition, Pr(B) equals the sum of the probabilities of the intersections of B and A1, A2,..., An.
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| SPECIAL CASE Law of Total Probability |
When S has only two partitions
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| De Morgan's Laws |
1. The complement of the union of n events is the intersection of the complements of the events
2. The complement of the intersection of n events is the union of the complements of the events
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| Conditional Probability | Is the probability that an event occurs based on a condition, or given that another event has occurred. Notation: the probability that event A occurs given that event B has occurred is Pr(A|B) |
| Conditional Probability Formula | Pr(A & B)/ Pr(A|B) = Pr (B) for Pr(B) not equal to 0 Similarly, the formula for A' given B is, Pr(A' & B)/ Pr(A'|B) = Pr(B) for Pr(B) not equal to 0 From the Law of a Total Probability, Pr(A|B) + Pr(A'|B) = 1 |
| Conditional Probability Independence Formulas | Pr(A&B) = Pr(A|B) • Pr(B) Formula conditioned on event A: Pr(B&A) = Pr(B|A) • Pr(A) |
| The Multiplicative Law of Probability | Since Pr(A&B) = Pr(B&A) Pr(A|B) • Pr(B) = Pr(B|A) • Pr(A) |
| Bayes' Theorem | Pr(Ai|B) = [Pr(B|Ai) • Pr(Ai)] / [SUM of Pr(B|Ai) • Pr(Ai)] |
| Factorial | Number of permutations of a set of n instinct elements Formula: n! ORDER MATTERS |
| Permutations | Number of ways to arrange k out of n elements n! / Formula: nPk = (n-k)! ORDER MATTERS |
| Combination | Number of ways to draw k elements out of n distinct elements n! / Formula: nCk = (n-k)!•k! ORDER DOES NOT MATTER |
| Partition | Number of ways to divide n distinct elements into r groups ORDER DOES NOT MATTER (more general than combination) ------------------------- Number of permutations of n non-distinct elements ORDER MATTERS (more general than factorial ------------------------- Formula: n! / k1•k2!•...•kr! |
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