Math 106 Lecture 3 Probability - Basic Terms Combinatorics and Probability - 1 Odds, Payoffs Rolling a die (virtually) m j winter, 00 1 Description We roll a six-sided die and look to see whether the face showing is divisible by 3. 1
Vocabulary Roll a (6-sided) die Experiment 1,, 3,,, 6 (Equally likely) Outcomes Face is divisible by 3 Event 3, 6 Outcomes favorable to the event Probability of event = P(face divisible by 3) =... number of (e.l.)outcomes favorable to the event total number of (e.l) outcomes... 1 = = 6 3 3 What is probability of exactly Heads in 10 tosses of a coin? Experiment: toss a coin 10 times, record the sequence of H and T H H H H H Count the outcomes: = 10 possible, equally likely, outcomes. Event: Exactly of the tosses are heads There are 10C ways to put down the H s. Others must be T s. p(exactly heads in 10 tosses) = 10C/ 10 = 0.6..
p(exactly heads in 10 tosses) = 10C/ 10 = 0.6 What does this mean? 0.6 0. = 1/ The chances are 1 out of that you will have exactly heads If you did the experiment a lot of times, about 1/ would be heads. If a lot of people did the experiment once, you d expect about 1/ to get exactly heads. What s the probability of exactly 10 heads in 0 tosses? Now there are 0 possible outcomes Of these, 0 C10 will be exactly 10 heads. 0C 10 = 0 0176.... Proportional Reasoning usually does not apply in probability. 6 3
Look at this with Excel 7 Roll Two Dice; sum the faces showing List the equally likely outcomes 1 3 6 1 3 6 7 3 6 3 6 7 8 6 7 8 9 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 1 Sample Space: set of all outcomes. Sometimes listed as equally likely, sometimes listed as distinct. 8
Sample Space for Sum of Two Dice Distinct Outcomes 11 1 3 10 9 8 6 7 Equally likely outcomes 3 3 9 1 3 6 1 3 6 7 3 6 7 8 3 6 7 8 9 Find the probability of each event 6 7 8 9 10 6 7 8 9 10 11 6 7 8 9 10 11 1 You roll doubles 6 1 36 = 6 0.16666... The sum is divisible by 9 1 = 36.... The sum is 6, 7, or 8 16 = 36 9.. 0.0.. 0... When comparing probabilities, use decimals 10
Combinatorics & Probability Your twin nieces received two Nancy Drew books for their birthday and would like some more. You don t know which ones they have, but Amazon.com has 1 Nancy Drew books. You decide on three at random. What is the probability all three are new to them? 11 Experiment: Select 3 from 1 Number of outcomes Nancy Drew - 1C3 Event: The 3 do not overlap with the they have. Number 1 3 of outcomes favorable to the event:.. P(no overlap) = 1C3 = 0.961... 1C3.. 1! = 96,90 3! 11! = 1! C = = 73,800 3!19! 1 6
Roll a die with your calculator TI: MATH ==> PRB ==>randint( ==>1,6) Display shows: randint(1,6) TI Roll three dice: randint(1,6,3) want 3 random integers between 1 and 6 13 Roll a die with Excel (this week s project) 1.Open Excel. If a new, blank spreadsheet doesn't appear, go to File, New, and click OK. Move to cell A1. Type =RANDBETWEEN(1,6) hit return. An integer between 1 and 6 should appear. If it doesn t: Go to Tools, Add-Ins, and put a check next to Analysis Toolpak 3. Select cell A1. Move the cursor until it changes to the + sign, then hold down the mouse button and drag down so that cells A1 through B36 are highlighted. The specified area should fill with random integers between 1 and 6... Hit F9 to re-evaluate the spreadsheet. A new set of random numbers will be generated. Instead of #3, you could use =INT(6*RAND()+1) 1 7
The formula: =INT(6*RAND()+1) RND() is a random number (not integer) between 0 and 1; never 0 and never 1. 6*RND(1) is a random number between 0 and 6. never 0 and never 6. 6*RND()+1 is a random integer between 1 and 7 (never 1 and never 7) INT(6*RND()+1) is an integer between 1 and 6, inclusive. 1 Back to Excel: Checking for Doubles Use the IF function; if the values are the same, we want a 1 ; otherwise a 0 Hit Enter, and drag down to D36 Find out how many 1 s in column D. (Add column D) 16 8
Rolling Doubles with Excel Could also use =SUM(D:D) or =SUM(select the range) 17 Odds Simple Game: roll a die. You win if 1 or ; you lose if 3,,, 6 P(win) = ; P(lose) = 6 6 To calculate your odds, (the odds of your winning), take: P(win) 6 1 P(lose) = = = 6 Your odds = odds of your winning are 1: Odds of your losing are :1 18 9
Odds and Vocabulary Suppose there are a equally likely outcomes favorable to event A, and b favorable to event B. Suppose that the total number of outcomes is a + b. a:b = odds that A occurs a:b = odds against B occurring Sometime phrased as odds of B are a:b against 19 Fair Payoff P(win) = ; P(lose) = Odds against you are - 1 6 6 Suppose you pay $1 to play each game. If you lose, you lose the dollar. In the long run, you will lose twice as often as you win.how much should you win if the game is to be fair? i.e., your average winnings are $0. Consider an ideal world. We ll play 6 games; you lose times, and you win times. You ve paid $6 to play. For the game to be fair, your winnings should total $6. You won twice, so your winnings each game should be 6/ = $3. You get your $1 back plus $ more. 0 10
Fair Game: Odds and Payoffs If the odds against you are - 1; the payoff should be - 1. This means, you get back your bet (stake) plus two times the amount of the stake. There will be more about fair games in a later lecture. 1 Problems with odds The odds of your winning are - against. What is the probability of winning? The odds against you are -. If the game costs $1 to play, what should the payoff be if the game is fair? 11
Problems with odds The odds are - against. What is the probability of winning? pw ( in) = 7 The odds against you are -. If the game costs $1 to play, what should the payoff be if the game is fair? Another way to express the odds is. =. 1 The payoff should be. - 1; i.e., you should receive $.0 plus your bet whenever you win. 3 Roulette Payoffs 1
Odds vs Payoff Payoffs Calculated as if no 0 and no 00 1 P(A) = ; odds of A 1:37 38 Odds against you are 37-1 Payoff is only 3-1 6 13