---start biostat 1.13.97------ I have absolutely NO idea what he's talking about, but he's writing this stuff on the board. AB BA BC CB AC CA (note- i'm not really using lowercase n to mean a different variable, it's just my way of making a subscript when this text editor doesn't let me make subscripts) nCr = N!/[r!(N-r)!] nPr = N!/(N-r)! see handout. so that's where we left off. poker: a game of chance sometimes played for money. deck of 52 cards, standard playing hand is 5 cards obtained one way or another. whoever has highest hand wins. low -->high pair two pair three of a kind straight (any suit; consecutive cards) flush (all one suit) full house (?) four of same card straight flush - same suit, consecutive cards calculate probability of a flush P(flush) = #flushes in each suit ---------------------- = total # 5 card hands N13C5 = 13!/[5!(8!)] = 1287 per suit 4 suits * 1287 = 5148 flushes total but, there's a slight glitch here. by calculating this way you're including the straight flushes. there are 9 straight flushes/suit = 36 total so this is the numerator. now, the denominator: 52 cards/deck, 5 cards/hand 52C5 = 52!/5!47!= 2598960 [oh god help me i have absolutely NO idea what is being discussed] so: P(F) = 4*13C5-36/52C5 = 5112/2.6 s 10to the 6th = .001967 or about .002 aka 2 out of a thousand. so getting a flush is rare! only about 2 chances in a thousand of getting this dealt to you. he remarks that it seems like they come more often than that, but apparently this is the probability. i wouldn't know. eg 2 4 6 8 10 of hearts is a flush, 2 3 4 5 6 of hearts is a straight flush. there are 36 total straight flushes possible. a quick question more in the veterinary area re: use of permutations equation...in horse racing, the first horse to finish - a trifecta is a rare probability event. when you bet you go up to the window you say you want horse # N to win place or show. but if you're really interested in rare bets you can bet a trifecta, where you say I want horse A B C to win, place, show in that order. assuming horses have equally likely chance of winning,which of course isn't the case, what is the chance of winning the trifecta? say there are 8 horses. 8P3 = N!/(N-r)! = 8!/5! = 336 permutations for 3 horses to finish in a particular order. so the probability is 1 out of 336, or about 3 out of a thousand, aka more likely than getting a flush. and since it's not totally random, you could make it better. nPk1k2...ki say you have 4 objects, two As and two Bs. how many permutations of these objects are there and how many ways can they be ordered given they fall into distinct subsets where within the subsets the orders can't be discerned. AABB ABAB ABBA BABA BAAB BBAA now if you had 8 heads and 4 tails the number of permutations increases. nPk1k2...ki for example above: 4P2,2 = 4!/2!2! = 6 this equation is mathematically equivalent to the binomial equation. however as we speak of binomial equation, (he says they got that in high school) (I don't know what it is, maybe if he tells us I will recognize it), in the clinical trials business, it has to do with random assignment. you should understand this.it is the basis of random clinical trials. consider what can happen with 3 tosses of a coin N tosses = 3 probability of heads = p(t) = 1/2 what we will delineate here is what can happen with three coin tosses 1 2 3 H H H 3H T H H 2H, T H T H H H T T T H 2T, H T H T H T T T T T 3T for any three tosses: p2H = 3/8 p3H = 1/8 p3T = 1/8 p2T = 3/8 1331 is supposed to remind us of something called pascal's triangle now, you can use the permutations equation to give this info 3P2,1 = 3 3P1,2 = 3 3P3,0 or 0,3 = 1 because these events are independent and equal, p any sequence of heads or tails is 1/2 cubed * n. P (x heads/n tosses) = nPx,(n-x) [1/2]^n so if it's 2 heads/3 tosses then 3P2,1 [1/2]^3 = 3*1/8=3/8 BUT if p(h) = 1/3 and p(t) = 2/3 then these outcomes are different. 3H 1 (1/3)^3=1/27 1/27 2H 3 (1/3)^2(2/3)=2/27 6/27 H 3 (1/3)(2/3)^2=4/27 12/27 T 1 (2/3)^3=8/27 8/27 P(x heads/n tosses) = nPx(n-x) * P(h)P(t) 3!/2!1! * (1/3)^2(2/3)^1 = 6/27 ----end------