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# INTRO TO PROBABILITY STA 4321

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This 18 page Class Notes was uploaded by Golden Bernhard on Friday September 18, 2015. The Class Notes belongs to STA 4321 at University of Florida taught by Arthur Berg in Fall. Since its upload, it has received 24 views. For similar materials see /class/206566/sta-4321-university-of-florida in Statistics at University of Florida.

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Date Created: 09/18/15

D o m an iple of Counting oun If the rst task has n1 passible autcames and the secand task has n2 passible autcames then there are mnz passible autcamesfar the twa tasks tagether Suppose the rst task is to roll a vesided bluecolored die and the second task is to role a sixsided orangecolored die How many different outcomes are there I rum39n mln M ml iple of Counting yr r V 7 lt oun If the rst task has n1 passible autcames and the secand task has n2 passible autcames then there are mnz passible autcamesfar the twa tasks tagether Suppose the rst task is to roll a vesided bluecolored die and the second task is to role a sixsided orangecolored die How many different outcomes are there Ans 6 5 30 I rum39nEmln M ml inciple of Counting 111 1111111 V If the rst task has n1 passible autcames and the secand task has n2 passible autcames then there are mnz passible autcamesfar the twa tasks tagether Suppose the rst task is to roll a vesided bluecolored die and the second task is to role a sixsided orangecolored die How many different outcomes are there Ans 6 5 30 1112131415 162122 2324 25 26313233 34353641 42 4344454651 5233343536 rmnlmh FiVSided Die Two questlons to ask o Is the order important 0 Is it with or without replacement 13 The number of ordered arrangements permutations of r objects selected from n distinct objects r g n is n Pfnn71nir1 nir n n1 n2 nr1 lchuice Ehoicesl Ichoicegl o o O Ehoicesl How many threeletter words can be made from the letters A B C D and E7 Here any combination of letters counts as a word V 111 Rem 111 The number of ordered arrangements permutations of r objects selected from n distinct objects r g n is n Prn n71n7r1 n 7r n1 n2 nr1 lchuice Ehoicesl Ichoicegl o o O Ehoicesl How many threeletter words can be made from the letters A B C D and E7 Ansng 543 Here any combination of letters counts as a word The number 0f distinct subsets cambinatians 0f size r that can be selected fram n distinct Objects r g n is given by Any permutatian 6le xr can be ardered in r di erent ways and s0 H fallaws Example Example 2 8 In Florida s lottery balls numbered from 1 to 53 are placed in a hopper and six balls are drawn at random without replacement What is the probability of winning this lottery I ruma mh The number 0f distinct subsets cambinatians 0f size r that can be selected fram n distinct Objects r g n is given by Any permutatian 6le xr can be ardered in r di erent ways and s0 H fallaws Example Example 2 8 In Florida s lottery balls numbered from 1 to 53 are placed in a hopper and six balls are drawn at random without replacement What is the probability of winning this lottery Ans 122957480 since 53 53 53 53x52x51x50x49x48 676I537676I47I7 6x5x4x3x2x1 72279577480 5 l ulnuh l mtg Formula Table 7 Figure 219 Counting the number of ways to select r items from 14 in tour different Order is Important Order is Not Important n quott1 PM 11 C77 n I7 I 7quot V r 71 rr r With Replacement Without Replacement Partit The number 0f ways afpartitianing n distinct abjects inta k graups cantaining n17 n2 Jik abjects respectively is n n13n2Xnkl Where 21 m n L 3921 1k quotk1 quot1quotnin1lgt n2IrinnHImlgt m n 7 n1 ink X O39I quot16703 Partitions Example Suppose ten employees are divided among three job assignments with three employees going to job A four to job B and three to job C In how many ways can the job assignments be made Partitions Example Suppose ten employees are divided among three job assignments with three employees going to job A four to job B and three to job C In how many ways can the job assignments be made Ans This can be accomplished in 10 10x9x8x7x6x5 7 4 200 3l4l3l 3 x 23 x 2 7 ways Birthday Problem Example Birthday Problem Suppose n people are in a room In this problem you may ignore the possibility of someone s birthday being on February 29 0 What is the probability that no two of them have the same birthday 9 Determine the smallest n such that the probability of at least two people have the same birthday is greater than 12 Birthday Problem Example Birthday Problem Suppose n people are in a room In this problem you may ignore the possibility of someone s birthday being on February 29 0 What is the probability that no two of them have the same birthday 9 Determine the smallest n such that the probability of at least two people have the same birthday is greater than 12 0 Ans There are 365 different outcomes The number of possibilities of everyone having a different birthday is 365364363 365 7 n 1 Therefore the probability is 365 365 7 nl365 forn 365 andO ifn gt 365 I l forn in 1 catn17prod365i0 1 2 3 4 5 6 7 8 9 1 1 100 002739726 008204166 01635591 02713557 04046248 0562357 07433529 09462383 1169482 1411414 1670248 1944103 2231025 15 02529013 16 0283604 17 03150077 18 0 469114 0 o o o o o o o o H o o o H H H b w M o o o n71365 nquotnquot 3791185 4114384 4436883 4756953 5072972 5383443 5686997 5982408 6268593 6544615 6809685 7063162 7304546 7533475 7749719 7953169 8143832 0 o o o o o o o o o o o o o o o o 0 8321821 848734 8640678 8782197 8912318 9031516 9140305 9239229 9328854 9409759 9482528 9547744 960598 9657796 9703736 974432 9780045 9811381 0 o o o o o o o o o o o o o o o 0 983877 9862623 9883324 9901225 991665 9929894 9941227 9950888 9959096 9966044 9971905 9976831 9980957 99844 9987264 9989637 9991596 Consider a wellshuf ed 52card deck of cards After being drawn 5 cards calculate the following probabilities royal ush AKQJ10 in the same suit straight ush straight in one suit four of a kind full house pair plus 3 of a kind straight sequential cards in different suits three of a kind two pair one pair 999009000 none of the above

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