8.2 Geometric Distributions Traits of the geometric setting: 1. Each observation has just 2 outcomes: a success and a failure. 2. The n observations
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1 8.2 Geometric Distributions Traits of the geometric setting: 1. Each observation has just 2 outcomes: a success and a failure. 2. The n observations are all independent. 3. The variable counts the number of trials needed to obtain the first success. (This was different in 8.1.) 4. The probability, p, of a success is the same for each observation. Geometric mnemonic: BIOS Binary just 2 outcomes: success or failure Independent trials (success probability doesn't change) (close enough if we pick <10% of population) Open ended number of trials Success probability is = for each trial before we begin 1
2 Ex. rock/paper/scissors in one game, someone will win or there's a tie RR RP RS PR PP PS SR SP SS If X = # of games played until there's a winner, then X is a geometric random variable with the probability of a success p = 6/9 = 2/3. 2
3 LET'S PLAY... Normal, Binomial, or Geometric? we record the # of defective bulbs in a box. we record the # of hours the bulbs last. we record the # of bulbs checked until we find a non defective bulb at the factory. we record the weight of each bulb. we record the number of bulbs that are not defective in a box. we record how many boxes must be checked until we find one in which both bulbs are defective. Binomial Normal Geometric Normal Binomial Geometric 3
4 geometric probability that the 1 st success occurs on the n th trial is P(X=n)=(1 p) n 1 p ex. We have a limitless supply of light bulbs. Suppose P(defective bulb) = 0.1. Apply the formula above to find:...the probability of a defective bulb on the first try....the probability that it takes 2 tries to find a defective bulb....the probability that it takes 4 tries to find a defective bulb. 2nd VARS D:geometpdf geometpdf(p,x) 4
5 ex. Apply the formula to find:...the probability that a family has a girl right away....the probability that a family has a boy then a girl....the probability that a family has all boys until having a girl as the 5th child....the probability that it takes 10 children for a family to finally have a girl. 5
6 Ex. Suppose 20% of cars contain jumper cables. Assuming independence, what's the probability I ask up to 3 people in order to find cables? not independent if I ask others who... are also looking for jumper cables who all came in the same car who all belong to the International Anti Jumper Cable Association p , 39 6
7 If X is a geometric random variable with probability of success p on each trial, then the mean, or expected value, of X is: μ=1/p The mean tells you the expected number of trials required to get the first success. (The variance formula is not an AP topic.) 7
8 Ex. What's the expected # of light bulbs inspected (p=0.1) until I find a defective one? Ex. If 20% of cars contain jumper cables, How many people you would expect to ask in order to find someone with cables? Ex. For rock/paper/scissors, what's the average # of rounds in order for one of them to win? 8
9 The probability that it takes more than n trials to see the first success is P(X>n)=(1 p) n Ex. For light bulbs, assuming p=0.1, what's the probability we inspect more than 4 bulbs to find our first defective bulb? What's the probability we inspect more than 12 bulbs to find a defective bulb? For the number of girls in a family, what is the probability that it takes more than 3 children before a couple has a girl? What is the probability that it takes more than 4 children before a couple has a girl? 9
10 Ex. If 20% of cars jumper cables, what's the probability that I have to ask more than 5 people to find cables? 10
11 For light bulbs, let p=0.1, the probability that I inspect up to 5 bulbs to find the first defective bulb: (the 1 st or the 2 nd or the 3 rd or the 4 th or the 5 th ), we add each of the geometric probabilities: P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5) This is the geometric cumulative distribution function on the calculator. 2 nd VARS E:geometcdf geometcdf(p,x) Ex. If 20% of cars have jumper cables, what's the probability I have to ask up to 4 people to find cables? 11
12 Ex. What's the probability it would take up to 5 tosses to get tails? Ex. What's the probability that I'd have to roll a die up to 5 times in order to roll a 6? 12
13 Binomial Distributions Geometric Distributions p , 43, 45 13
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