hapter () Discrete robability Distributios Eamples Eample () Two balaced dice are rolled. Let X be the sum of the two dice. Obtai the probability distributio of X. Solutio Whe the two balaced dice are rolled, there are 3 equally likely possible outcomes as show below: (,) (,) (3,) S (,) (,) (,) (,) (,) (3,) (,) (,) (,) (,3) (,3) (3,3) (,3) (,3) (,3) (,) (,) (3,) (,) (,) (,) (,) (,) (3,) (,) (,) (,) (,) (,) (3,) (,) (,) (,) The possible values of X are:, 3,,,, 7, 8, 9,, ad. The discrete probability distributio of X is give by X (X) / 3 3 /3 3/3 /3 /3 7 /3 8 / 3 9 /3 3/3 /3 / 3 Total 3/3 =
Eample () The umber of persos X, i Al Riyadh family chose at radom has the followig probability distributio: X 3 7 8 Total (X).3....... / Fid the average family size { E(X)} / Fid the variace of probability distributio Solutio X (X) X(X) *.3.3 -.....88 -... 3..33.9.8.89...9 3.....9 8..8.. 3.9.. 7..7.9.. 8..8.9 3.8.38..3 ( ) E =. = 3 perso =.3 Eample (3) Joh Ragsdale sells ew cars for elica Ford. Joh usually sells the largest umber of cars o Saturday. He has developed the followig probability distributio for the umber of cars he epects to sell o a particular Saturday. Total p...3 3.3. p( ) Fid:. Epected value of (The mea of probability distributio ). (The variace of probability distributio )
Solutio: ( ) E 3 p...3.3. ()...9.. -. -. -..9.9....8 3....3.3.3.9 p...3 3.3.. ( ) E......3.8 3 3....3.3.3.9 Eample () Which oe of these tables is actually a probability distributio? 3
Biomial distributio Eample () If the eperimet is tossig a coi times, what is the probability of:. Gettig two heads.. Gettig at least heads. 3. Gettig at most oe head.. Gettig at least two heads. Fid mea, variace ad deviatio Solutio:!!!, =,,,..3 ()! ()()!!!!..3!!! 3!!!!!! 3.. 7. 89.. p. 3... Variace
Eample () Over a log period of time it has bee observed that a give marksma ca hit a target o a sigle trial with probability equal to.8. Suppose he fires four shots at the target. Aswer the followig:. What is the probability that he will hit the target eactly two times?. What is the probability that he will hit the target at least oce? 3. Fid mea, variace ad stadard deviatio. Solutio:.8.8...8..!!......8.!.8.!.8..!!.. 99 3..8 3..8....8 Eample (7) Fid the probability of guessig correctly eactly of the aswers o a true-false eamiatio. Solutio:.
Hypergeometric Distributio Eample (8) Horwege Discout Brokers plas to hire ew fiacial aalysts this year. There is a pool of approved applicats, ad George Horwege, the ower, decides to radomly select those who will be hired.there are 8 me ad wome amog the approved applicats. What is the probability that 3 of the hired are me? Solutio: N,. N 8.7., S 8, X 3 p s X N S X 3 N 8 3 8 3. Eample (9) Suppose we radomly select cards without replacemet from a ordiary deck of playig cards. - What is the probability of gettig eactly red cards (i.e., hearts or diamods)? - What is the probability of obtaiig or fewer hearts? Solutio: - This is a hypergeometric eperimet i which we kow the followig: N = ; sice there are cards i a deck. S = ; sice there are red cards i a deck. = ; sice we radomly select cards from the deck. = ; sice of the cards we select are red. We plug these values ito the hypergeometric formula as follows: p s X N S X N.3 Thus, the probability of radomly selectig red cards is.3.
- N = ; sice there are cards i a deck. S = 3; sice there are 3 hearts i a deck. = ; sice we radomly select cards from the deck. = to ; sice our selectio icludes,, or hearts. oisso distributio Eample () The average umber of traffic accidets o a certai sectio of highway is two per week assume that the umber of accidets follows a oisso distributio with.. Fid the probability of o accidets o this sectio of highway durig a - week period.. Fid the probability of at most three accidets o this sectio of highway durig a -week. 3. Fid the probability of at least four accidets durig a -week. Fid variace ad stadard deviatio. Solutio:..78 X e.3!!. 3 3 3.78.78.78.78!!! 3!.873 3. 3.873. 7 7
. Var. Eample () Suppose a life isurace compay isures the lives of 3 me aged. If actuarial studies show the probability that a -year-old ma will die i a give year to be o.oo, fid the eact probability that the compay will have to pay = claims durig a give year. Solutio: u 3. 3 3 3.78.8! Eample () The umber of people arrivig to a movie theater i a 3-miute time period is best modeled usig which of the followig distributios? A) Normal B) Biomial ) Hypergeometric D) oisso A maufacturig process produces defective items % of the time. A sample of items is take for quality cotrol. If you wated to determie the probability that eactly of the items are defective, which distributio should be used? A) Uiform B) Biomial ) Hypergeometric D) oisso Which of the followig is ot a requiremet of a biomial distributio? A) A costat probability of success. B) Oly two possible outcomes. ) A fied umber of trials. D) Equally likely outcomes 8