Geometric Distribution & Poisson Distribution Week # 8 Geometric Distribution: • A trial is repeated until a success occurs. • The repeated trials are independent of each other. • The probability, p, is constant for each trial Formula: x 1 P( x) p(q) , where: q 1 p When it is used: Ex (1) – Prizes are being awarded in a raffle. The tickets get put back into the hat each time there is a drawing. Your name is placed in a hat with nine others. Find the probability that your name will be pulled on the 5th draw? 51 P(5) (1/10)(9 /10) P( x) .06561 So what happened in Ex (1) • We were looking for a specific time for our event to happen: Trial # 5 • So, we wanted 1 success and 4 failures 51 P(5) (1/10)(9 /10) Use on the TI-83 • • • • • 2nd DISTR Scroll down to Geometpdf( Put in the P(E),number of trials Geometpdf(prob,# of trials) Hit enter! Tom Brady Tom Brady completes 65% of his pass attempts. Find the following: (a)What is the probability he will complete a pass on his 5th throw? (b)What is the probability of him finally completing a pass on his 10th throw? Let’s try one more….. – Coin toss. a: P(heads) on the first toss? b: P(heads two times in a row)? c: P(the first heads that happens is on the third toss)? d: P(heads will occur on the 5th or 6th toss)? e: P(the first heads occurs on the 37th toss)? Poisson Distribution: • Counts the number of times an event occurs in a given interval, you must know or be given the MEAN (μ). • The Probability is the same for each event. • The number of occurrences in one interval is INdependent of the number of occurrences in another window. x Formula: e P( x) x! Let’s try one…. Ex (3) – The average number of days of sun in the month of December in Buffalo is 12 days. What is the probability that next December there will be 15 days? P( x) e x x! 15 12 12 e P( x) 15! P( x) .0724 Extending: • Let’s do PRB # 22 from p. 227…… A newspaper finds that the mean number of typographical errors per page is 4. (a)Find the probability that there are going to be 3? (b)Find the probability that there are going to be 7? Using the TI-83 2nd DISTR Scroll down to Poissonpdf( Type in the mean(μ), x value. Poissonpdf(12,15) for the last problem. You must be able to….. • determine when to use: – Binomial Distribution – Geometric Distribution – Poisson Distribution • See page 225…..