EXAM 1 (notes).pdf

Basic probability probability (P) = favorable outcome total outcome : steps i s 1) identify total # of outcomes 2) identify # of favorable...

Basic probability probability (P) = favorable outcome total outcome : steps i s 1) identify total # of outcomes 2) identify # of favorable outcomes 3) use probability formula 4) simplify fraction Ex I. A deck of 52 cards is shuffled + one card is drawn & random. What is the probability that card is a heart decks piheart) = 13 heart 52 total curds Basic Set Operation P(AUB) = P(A) + P(B) - P(A1B) SetA SetB union (U) : probability that element A or B or both ↳ set A (only ↳ Set B (only ↑ ↳ both set A & B (overlap Set AUB intersection (1) : probability that element is in both A and B compliment : probability that element is not in a given set steps : ~ 1) identify P(A) , PCB) , and PLANB) 2) convert given # into probabilities 3) plug into formula EX.2 In a class of 100 students , 60 are enrolled in Mathematics (Set A) , 50 in Physics (Set B) , and 30 in both courses What. is the probability that a rand- only selected student is enrolled in Mathematics or physics P(AUB) = P(A) + P(B) P(AB) - P(A) = 0 6. P(AUB) = 0 6. + 0 5. - 0 3. = 0 8. p(B) = 0 5. P(AnB) = 0 3. Mutually Exclusive Events events A & B are mutually exclusive if AB = 0 ↳ cannot happena same time P(AUB) P(A) + P(B) = Ex 3. Suppose 2 events A & B are mutually exclusive. P(A) = 0 3. and PCB) = 0 4.. What is the probability of AUB ? P(A1B) = 0 P(AUB) P(A) + P(B) = = 0 3+0 4... = 0. 7 Independence events A & B are independent ; the occurrence of one does not effect the probability of other P(AB) P(A) X P(B)= Total Probability Rule call the probability of an even , PCD)-broken dwn by diff condition. ↳ involves mutually exclusive events conditional probability conditional probability of D of D Boccurs given that A occurs given that - & x = PETA)XA) + P(D(B) X P(B) u probability probability probability of event D that event A of event B occurring occures occurring Conditional Probability P(AIB) P(ANB) = or P(BIA)P(A) P(B) P(B) probability of event occurring given that another event has alr occurred. PCAIB) probability of A given B PLANB) probability both A & B occure P(B) probability that B occurs * this is only defined when P(B) > O meaning event B must have a non-zero probability Counting Questions (Combinatorics) asked to count # of ways certain events or arrangements can occur ↳ 4 diff types : 1. permutation : arrangement order matters P(n r) n! , = (n-r) ! 2. Combinations : order does not matter ((n , r)= n ! r! (n r) ! -. factorial 3 : (n ! ) involve multiplying integers up to n! 4. multiplication principle : if , event occurs in m ways a another event both event occur is occurs n ways , than totalIt of ways mxn Random variable cumulative distribution function (CDF) : probability that random variable will take a value less than or equal to a certain value E(X) =) cumulative probability random variable X probability takes on a value less than up to X or equal to Discrete random variable : finite numbers/countable # Of possible outcomes listed # of sum of probabilities of all outcomes less than or equal to %. If X is discrete random CDF is : probability that random variable X takes place less than or equal to certain value X I F(x) P(XIX) (x xi) = = = random variable Sum of probabilities P(X Xi) = for all Xi add up all values of X that are