Basic things you need to know about sets and probability Symbol or { } U ' ∪ ⋂ Name Typical Example Trickier Example empty set { } ={humans on Mars} (True) = { } (False) is an element of 2 {1, 2, 3} (True) not an element of 2 {1, 2, 3} (False) is a subset of {a, b} {a, b, c} (True) {a, b, c} {a, b, c} (True) a proper subset of {a, b} {a, b, c} (True) {a, b, c} {a, b, c} (False) universal set complement If U={1, 2, 3} & A={1}, A'={2,3} union If A={p,q}, B={r,s}, A∪B={p,q,r,s} intersection If A={p,q,r}, B={q,r,s}, A⋂B={q,r} Two sets are equal if and only if they have the same elements (and order does not Other comments {} is a subset of every set every set is a subset of itself no set is a proper subset of itself the set of all elements possible must know U to find complement matter). Ex: {1, 3, 5, 7} = {3, 1, 7, 5} but {1, 3, 5} ≠ {1, 3, 7} A set with n elements has Union Rule for Sets: Union Rule for Probability: P(A∪B) = P(A) + P(B) – P(A⋂B) Complement Rule: P(E) = 1 – P(E') and P(E') = 1 – P(E) Basic Probability Principle: If E is an event and S is its equally likely sample space, then subsets. n(A∪B) = n(A) + n(B) – n(A⋂B) ( ) ( ) ( ) ( ) Odds: A practical way of calculating odds and how odds relate to probability: ( ) Think of the sample space as being divided into two parts… o The things that you want to happen…call this “The part you want” and o The things that you don’t want to happen (everything else)”…call this “The part you don’t want” o So, n(part you want) + n(part you don’t want”) = n(WHOLE sample space) (You don’t literally have to want these events to happen…it’s just a nice way of phrasing it in my opinion.) o From this point of view, odds are always calculated using the “parts”: o To find the odds against, just flip the numbers in the odds in favor calculation o While odds are always “part to part”, probability is always “part to whole”: ( ) ( ( ) ) ( ( ) ( ) ) ( ( ) ) You need to know how to write out the sample space for a problem involving rolling 2 dice: 11 21 31 41 51 61 Conditional Probability: o 12 22 32 42 52 62 ( ) 13 23 33 43 53 63 ( 14 24 34 44 54 64 15 25 35 45 55 65 16 26 36 46 56 66 ) ( ) A practical way of calculating conditional probabilities: Think of the “condition” as something that REDUCES the original sample space and once the REDUCED sample space is determined, the conditional probability can be calculated as usual by simply using the Basic Probability Principle and the REDUCED sample space in place of the original sample space. Product Rule of Probability: P(A⋂B) = P(A) · P(B A) [ or, if you prefer: P(A⋂B) = P(B) · P(A Note: If A and B are independent events, then this simply becomes : P(A⋂B) = P(A) · P( B) The Product Rule for Probability can perhaps best be visualized via a tree diagram: Tree diagram illustrating Product Rule of Probability for Dependent Events. Tree diagram illustrating Product Rule of Probability for Independent Events. P(A⋂B) = P(A)·P(B A) (multiplying down the appropriate branch of the tree) P(A⋂B) = P(A)·P(B) (multiplying down the appropriate branch of the tree) B) ] Sets and Venn Diagrams A A⋂B A∪B A⋂B⋂C A⋂B' [or A-B] A' (A⋂B)' [or A'∪B'] (A∪B)' [or A'⋂B'] A'⋂B'⋂C' [or (A∪B ∪C)'] B⋂A' [or B-A] You may have noticed that when you complement a union or intersection, each set gets complemented and the operation switches (union becomes intersection and intersection becomes union). For example, (A∪B)' = A'⋂B' (A∪B ∪C)' = A'⋂B'⋂C' (A⋂B)' = A'∪B' (A∪B ∪C)' = A'⋂B'⋂C' So, if you need a Venn representation of one of these, use the one that seems easiest.