Probability Distributions. Probability Distributions. J. Boulton. May 08, 2013 MDM 4U1. Where are we?

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1 May 08, 203 robability Distributions robability Distributions The Distribution Binomial Geometric Hypergeometric Using Ecel Advanced applications The Distribution Binomial Geometric Hypergeometric Using Ecel Advanced applications rolling three 's in a row with a si-sided die? What if you didn't know what order they could occur in? rolling three 's with seven? tosses of a si-sided die? Who might want to know the answer to these questions, and why? How likely is it to have more than one defective part in this product package? How many doses are most likely to be sufficient to cure this patient? How much can we epect it to cost? How much do we have to charge for this lottery ticket to make a profit? What's the likelihood of more 9 calls coming in then we have ambulances for? Where are we? Conceptualizing a "Distribution" The Distribution Binomial Geometric Hypergeometric Using Ecel Advanced applications rolling three 's in a row with a si-sided die? rolling three 's with seven? tosses of a si-sided die? What's different between these two questions; what's the same? Is three fives in seven tosses the only thing that is possible? What else could happen? This is a discrete probability distribution. Any whole number of successes is possible. However, the likelihood varies with the probability of success (). If the probability of success were to rise, what do you think would happen to the appearance of this graph? Why? MDM 4U

2 May 08, 203 Achtung! Conceptualizing a "Distribution" The Distribution Binomial Geometric Hypergeometric Using Ecel Advanced applications drawing out 2 blue then green marble from this bag (with replacement)? robability Calculation (Using techniques from unit two) Can you etend the solution to this problem on the right? rolling three 's with seven? tosses of a si-sided die? drawing out 2 blue and green marble from this bag in any order (with replacement)? Note: there are 3 3 = 4 7.4% 3 C 2 = ways this could happen = 2.4% Conceptualizing a "Distribution" The Distribution Binomial Geometric Hypergeometric Using Ecel Advanced applications Can you come up with a formula for this kind of problem, and its conditions for use? Hint for conditions: what was significant about the way in which we drew out the marbles and how is that like tossing a die? rolling three 's with seven? tosses of a si-sided die? Solution rolling three 's in a row with a si-sided die? Here's another way: rolling three 's with seven? tosses of a si-sided die?... woah, now what? But, did it have to happen in this order? () = n C r r n-r How many more are there? Does this seem familiar to you? MDM 4U

3 May 08, 203? rolling three 's with seven tosses of a si-sided die? Isn't this just permutations of identical items (i.e. combinations)? How? These are probabilities of success and failure on each individual trial... robability of Success Can you have anything besides a or not a? (A) (A) + (A ' ) = robability of Failure (A ' ) = - (A)? rolling three 's with seven tosses of a si-sided die? To arrange these identical items, we're just permuting the letters p and (success and failure). 7! 3! 4! = n! r!(n-r)! = To arrange these identical items, we're just choosing trials to be successes, and trials to be failures. n C r = = 3 7 C 3 That means there are 3 arrangements of 3 successes and 4 failures (in seven trials).? rolling three 's with seven tosses of a si-sided die? But wait, we still don't know the probability of 3 successes happening! Binomial robability: Warm up. The faces of a 2-sided die are numbered from to 2. rolling 9 at least twice in ten tries? We know this: 3 arrangements And we know this: = 7.8% 2. A coin is tossed ten times. Find the probability that a) eactly four heads are tossed b) at least two heads are tossed c) no more than two tails are tossed Or, in general... () = n C r r this is the # n-r 3. In a multiple choice test that contains ten questions with each question having five possible answers, what is the probability that a) Colin will pass the test if he merely guesses at each question? b) Diane will get an "A" on the the test if she has studied and she feels that her probability of answering each question correctly is 0.7? 4. Assuming that the chance of giving birth to a girl or boy is even, what are the chances that a) a couple planning to have three children will have all girls? b) a couple planning to have five children will have at least one girl? MDM 4U

4 May 08, 203 What is a distribution Epected Value The Distribution Binomial Geometric Hypergeometric Using Ecel Advanced applications Epected Outcomes: Successes vs. Values Epected Outcomes: Successes vs. Values Successes A simple eample: How many times should you should have success in a certain number of trials? Just multiply the chance of success each trial, times the number of trials. E(X) = n Note, this formula is not always used for all distributions (they each have their own).. What if you toss a coin 0 times, how many 'heads' would you epect to get? Values i) a single outcome What is the value outcome most likely to occur in a trial? Just multiply the chance of success each trial, times the value resulting from a success (this is the success's proportion of the value per trial). A simple eample: E() = () Note, this formula is always used for all distributions (the last one is not). You just find () calculated using the appropriate formula for the distribution. a) If a jackpot is $ million, and your probability of winning is 0.00%, what is your epected winnings from purchasing a ticket? b) What would the lottery have to charge to break even with odds like this? MDM 4U

5 May 08, 203 Epected Outcomes: Successes vs. Values Values ii) multiple outcome What is the value outcome most likely to occur in a trial when more than one value outcome is possible? Just use the formula multiple times and add all the results together. Just multiply the chance of success each trial, times the value resulting from a success (this is the success's proportion of the value per trial). A simple eample: E() = n i = X i ( i ) Note, this formula is always used for all distributions (epected successes is not). () is calculated using the appropriate formula. 7a) If you earn $ each time a coin toss shows a head, and you must pay $2 each time it's a tail, what are your epected winnings/losses in ten tosses? 7b) What is the most you would pay to play this game? Binomial Distribution (Eercises) 8. Assume that every time uinlan, a hockey player, gets a breakaway on the opposition's net, he has a probability of 0. of scoring. If he averages two breakaways a game, what is the epected number of goals that he will score on breakaways in a season with 7 games? 9. In a manufacturing process, it is estimated that only 2 percent of the bolts that are machined are declared defective, that is, they are either too large or too small. In a package of 0 bolts, what is the probability that there is at least one defective bolt? How many would you epect? 0. If the probability is 0. that Luciana will hit a bull's eye on a dart board, what is the probability that she will get at least one bull's eye in ten attempts? How many would you epect her to get?. It seems that every carton of eggs at the supermarket contains at least one broken egg. If, in fact, it has been determined that 3 percent of the eggs supplied to a supermarket are cracked, what is the probability that if you buy two dozen eggs none of your eggs will be cracked? How many would you epect to be cracked? 2. Find the probability that at least three students in a class of 30 students were born on a Saturday. How many would you epect to be born on a Saturday in the class? 3. At the height of The Beatles' popularity, it was estimated that their music was played on every popular music radio station 40 percent of the time. What is the probability that if you tuned through ten such stations at any given moment at least one of the stations would be playing a Beatles song? 4. What is the relationship of the probability of a specific event, and the event's "proimity" to the epected value? Epected Value More comple eamples:. What is the epected sum of ten rolls of a single, si-sided die?. Your brother bets you that you can't roll more than 8 sies in twenty rolls of a die. If you do, he'll pay you $ per si. If you don't he wants to know what you'll pay him. What is the most you should offer? 7. What is the epected sum of ten rolls of two si-sided dice? 8. You and the dealer are playing blackjack with a brand new, well-shuffled deck. You have an ace, and the dealer has a seven. If blackjack pays $00, what is the most you should be willing to pay to receive the net card? Answer Clues ) 20% 2a) 20.% b) 98.9% c).% 3a) 3.3% b) 2.% 4a) 2.% b) 9.9% ) five a) $0 b) $0 per ticket 7a) $ won b) $.0 a toss 8) 90 9) 3.%/you'd epect one 0) 80.3%/you'd epect one ) 48.%/ you'd epect less than one cracked egg 2) 82.3%/~4.2 students 3) 99.4% 4) closer = greater ) 3 ) no more than 4 per si under 9 sies(epect to receive about 3 total) 7) 70 8) $32 MDM 4U

6 May 08, 203 aul the Octopus What are the chances? MDM 4U