CLPRPBKY Classical Probability

8/13/2019 CLPRPBKY Classical Probability

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SOME CLASSIC OR COMMON PROBABILITY PROBLEMS

1. (Banach's matchbox problem). A man buys two boxes of matches, each of whichcontains matches and puts them in his pocket. Every time he has to light a match,he selects at random one box or the other. What is the probability that there areexactly (1 matches left in one box when he lights the last match from the 5 8other box?

Let denote the probability we wish to find. We first look at the special case when a: 8 $5we will call the boxes and To find , we will find the probability given by, , : " # "T , , , after 4 lights, 2 matches used from 2 matches used from and last match used from" # "and then double it (to handle the opposite case where the last match came from , etc.).,#

Hence In a similar fashion, we find .: # : # " #% $# # $# ) " # "' )

" "# $ " ' $ & %

Finally, we get : # $#! # )

" # $

Conjecture : # 5#85"

8" "##85

Proof. The probability is double the probability of the event that, after lights,: #8 55the last match used was from and previously matches were used from and matche, 8 " , 8 5" "were used from . Hence, from the initial lights, we must choose the lights that ca, #8 5 " 8 "#from matches in There are possible choices (sequences). Therefore, "

#85"8"

number of sequences probability of a particular sequence: # 5

and thus

2: 5#85"

8" "

##85


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2. (Birthday problem). In random group of people, what is the probability that no twoof them have the same birthday (disregarding the year)? [Assume that birthdays are uniformlydistributed throughout the year and that none of the people are born on February 29.]

(no birthday matches)T number of permutations of items from 365 possible items

number of n-tuples in sample space

8

$'& 88T

$'&365 364 365365 365

8"$'&

3. (The matching problem). An absent-minded professor wrote 10 letters and sealed them inenvelopes without writing the addresses on the envelopes. Having forgotten which letter he hadput into which envelope, he wrote the ten addresses on the envelopes at random. What is theprobability that none of the letters was addressed correctly?

Answer. 12! 10 /" "$x %x x" "10

368

We solve the problem in general for letters.8

For let denote the event that envelope gets the correct address.3 " # 8 Q 33

Hence we want to find Using DeMorgan's law, the probability of a complement, andT Q 3 "

8+ 3probability of a general union, we have

1T Q T Q T Q

3 " 3 " 3 "

8 8 8+ - -3 3 3

" T Q T Q Q T Q Q Q " T Q Q 8! ! !

3" 343 3 4 3 4 5 " 8

345

8

" " 8 8 8 8" 8 # 88" $ 88"8# 8 8x" " " "8

" " "" " "#x $x 8x8


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4. (A Bayes' Theorem problem). Suppose a diagnostic test for a certain disease is 90% accuratIn other words the false positive rate and the false negative rate are both 10%.If 3% of the population has the disease, find the probability that a random personwho tests positive for the disease actually has the disease.

Let denote the event a random person has the disease, and let denote theH Xevent a random person tests positive for the disease. Then

T HlX #"((T H X T X lHT H *!!$

T X *!!$"!*(TX lHTHTX lHTH

5. (A hypergeometric probability problem). Suppose a shipment of 100 items contains 10defective items and 90 non-defective items. If 5 items are randomly selected from theshipment, what is the probability that exactly one of the 5 is defective?

Let denote the number of defectives in the sample. Then\

.3394T \ " 10 901 4

1005

6. (A binomial probability problem.) If a student randomly guesses answers on a 10question multiple-choice test which has 5 choices for each question, find theprobability that the student gets exactly 3 of the 10 correct.

Let denote the number of correct answers. Then\

T \ $ #!"$ "!$ & &" %$ (


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7. (Gambler's ruin problem.) Consider a gambler who at each play of the game hasprobability .45 of winning one dollar and probability .55 of losing one dollar. ; Assuming that successive plays of the game are independent, what is the probability that,starting with $10 dollars, the gambler's fortune will reach $15 before going broke?

answer:"

"

.55 .45

.55 .45

10

15 .3338

Let denote the probabilitythat she wins 15 dollarsT 5 5For example, denotes the probability that she is up 1 dollar before going brokeT"%Conditioning on the first play, we have .45 .55 Since we can writeT T T 10 11 9T T T 10 10 10.45 .55 , we obtain

.45 .55T T T T 11 10 10 9

which implies T T T T 11 10 10 9

.55

.45

Letting we have the recursive relationshipH T T 5 5 5"

for withH H 5 " # H T 5" 5 " "&&%&

Hence

H T 8 "&&%&

8"

Now note that T T T T T T 8 " # " 8 8"

H T ! ! 5" 5"

8 8

5 "&&%&

5"

T ""&&%&"&&%&

8

Furthermore, since the probability she will reach $0 winnings before going broke is tT ""&

probability of her status before the first play of the game. Hence we have T ""&&%&

"&&%&"&

Therefore,

T "! "&&%& "&&%&"&&%& "&&%&"&10

="&&%&

"&&%&

"!

"&

.3338


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8. (Coupon collecting problem). There are 4 different types of coupons: and X X X # $ %Each time a person collects a coupon the outcome is independent of other times and theprobabilities of getting the different types are andX T X # T X $# $TX % R. Let denote the number of coupons one needs to collect in order to have atleast one coupon of each type. Find and .R 8 IR

Let denote the event of having at least one coupon of type after collectingI 3 83coupons. Then T R 8 TI I I I " T I I I I " # $ % " # $ %

" T I - 3

" T I T I I T I I I T I I I I ! ! !3 4 4 5 # $34 3 3 " %345

" * ) ( ' ( ' & & % $ 8 8 8 8 8 8 8 8 8 8

% $ # " !8 8 8 8

" * ) #& # "8 8 8 88

Hence T R 8 J8 J 8 " " * ) #& # " 8 8 8 88

" * ) #& # " 8" 8" 8" 8"8"

"* #) )# *" &8" 8" 8" 8" 8"

IR 8T R 8 8"* #) )# *" & ! !8% 8%

_ _8" 8" 8" 8" 8"

8"* #) )# *" & !8"

_8" 8" 8" 8" 8"

8"* #) )# *" & !8"

$8" 8" 8" 8" 8"

" # ) * &" " " " #

" # ) * " #!* #"' #"' #!* " $!)" $"#) $!$# $!!* (&

"#$'""""""


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9. (A geometric probability problem). A pair of dice is rolled until a double-six occurs.Let denote the number of rolls. Find andTR 8 IR

T R 8 8 " # $ $& "$' $'8"

for

IR 8 8 $' $'! ! a b8" 8"

_ _$& " " $& "$' $' $' $' $'

8" 8" #

10. (A placement problem). A fair coin is flipped 10 times. Given that 3 headsoccurred, find the probability that no two consecutive heads occurred.

T no heads in a row # outcomes with no heads in a rowtotal number of outcomes

)$

"!$

) ( ' ("! * ) "&

To see why the number of outcomes with no heads in a row is , note

8

3that an outcome with no heads in a row is of the form

___ HT____HT____H____

where the remaining 5 tails are distributed among the 4 blanks above, which can be done

in ways. &%"%" 83A fair coin is flipped times. Given that heads occurred, find the probability that5no two consecutive heads occurred.

T no heads in a row 855"5

585

85"

585

T /


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11. (A Fibonacci-sequence related problem). A fair coin is flipped until two consecutiveheads occur. Let denote the number of flips. Find andIR T R 8

To find , we condition on the first flip resultIR IR IR lX IR lL" "# #

Note since a tail on the first flip means you have used one flip and you startIRlX " IRfresh. To find , we condition on the second flipIR lL

IRlL IRlLL IRlLX # # IR " " " "# # # #

Hence ,IR " IR # # IR IR " " " " $ $# # # # # %a b

which implies .R 6 To find , we look at the first few cases to see a pattern.T R 8

T R # "##

T R $ TXLL "#$

T R % T LX LL T X XLL # J J " "# #% %

$ 5where is defined below]

T R & T L X X L L T X L X X L T X X X L L $ J "# #&

%" %

T R 8 T 8 $ T X LLno heads in a row through flips

using result from problem 10] ! 5!

8$8$

5 # #" "8$ $

8$5"

58$

5

!"# 58

5!

8$85#

sincewe need for 5 8 5 # ! !"# 5 58

5!

85# 85#

8##

# 8 8where denotes the th Fibonacci number given by or defined recursively byJ 3 J 3 3

5

3"5

5! =0

[ 312

J J J J J " J 3# 3 3" " #

&

& "#with . Also, where3

3 39 9

9

For example, andJ & J && &% $ #! " #

& 9 9& &


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Other route to IR

IR 8T 8 8 8 J ! ! 8# 8#

_ _"#

88"

8! 8#

_"#

8

&

9 98" 8"

8 8 " " " "& &

8# 8#

_ _

# #

8 88" 8" ! ! 9 9

8 8" " "& &

"

8# 8#

_ _

# #

8 8

9 9! ! 9 9

8 8 " " " " "& # & & # &

"

8" 8"

_ _

# #8 8 9 9! ! 9 9

" "& &

" # "#

" " 9 99 9

9

9## "

##

# " " " " "# & # &

# ## a b9 9

'

12. (De Mere's problem.) Find the probability of getting at least one six in 4 rolls of a fair die.Compare to the probability of getting at least one double-six in 24 rolls of a pair of dice.

T " &"((&at least one six in 4 rolls of a fair die & '("' "#*'

%

at least one double-six in 24 rolls)T " %*"%! $&$'#%


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13. (The car and 2 goats problem). Randomly placed behind 3 doors are a car and 2 goats.After randomly picking a door, one of the other doors is opened (either randomly or bynecessity) to reveal a goat. You can stick with your original door pick or you can switch tothe remaining unopened door. Which strategy stick or switch provides the best chanceof picking the car?

The stick strategy provides a probability of for picking the car."$ The switch strategy provides a probability of 2 for picking the car since whenever$you originally pick a goat door, the switch will be to the car door.

14. (Winning at craps). The player throws two dice, and if the sum is 7 or 11, thenshe wins. If the sum is 2, 3, or 12, then she loses. If the sum is anything else, then shecontinues throwing until she either throws that number again (in which case shewins) or she throws a 7 (in which case she loses). Calculate the probability that theplayer wins.

T T T Win Win with 7 or 11 Win with 4 or 10 Win with 5 or 9 Win with 6 or 8 T T

Win with 7 or 11 2 Win with 4 T T

2 Win with 5 2 Win with 6 T T

1. Win with 7 or 11T 6 2 836 36 36

+

2. Win with a 4 Win with 4 on th rollT T 3 ! ! 3

_ _

=2 i=2

3 2736 36

2 i 2

3 1 3 36 136 1 27/36 36 9 36

2 2

3. Win with a 5 Win with 5 on th rollT T 3 ! ! 3 3

_ _

=2 =2

4 26 236 36 45

2 i 2

4. Win with a 6 Win with 6 on ith rollT T ! ! 3

_ _

=2 i=2

5 25 2536 36 36(11)

2 i 2

Therefore, Win 2( ) .4929T 8 1 2 25 244

36 36 45 36(11) 495


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15.(The forgotten digit problem). You forget the last digit of a friend's phonenumber. You randomly pick a digit and dial. Let denote the number of attemptsuntil you dial correctly. Find for .R 5 5 # 8

T R 5 5 " 8* ) "!5" " ""! * "!5# "!5" "!

for

16. (A lottery problem). Six different numbers from 1 through 49 are randomlypicked. Find the probability of guessing all 6 numbers on one try.

guessing all 6T " ""$*)$)"' %*

'

17. (The gold coins problem). There are two boxes. One box contains two goldcoins. The other box contains one gold coin and one silver coin. You randomly chosea box, then blindly pick a coin. Given that the coin picked is gold, find theprobability that the box you chose was the two-gold coin box.

T # l gold pick gold T "#

T "#""#"# $#pick 2 gold box

first coin picked is gold


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18. Deal five cards from a well-shuffled deck of cards. Find the probabilities for thefollowing hands: a pair, two pairs, three-of-a-kind, a full house, four-of-a-kind, a plainstraight, a plain flush, a (non-royal) straight flush, a royal flush, and a junk hand.

" T 06?=2 !!!!"$)&"'*% * %" "

&

$'#&*)*'!

3. - - -T 0 9?< 90 + 538. !!!#%!!*'!$)%

"$ %)" "

&

'#%#&*)*'!

4. T 0 ?66 29?=/ !!"%%!&('# "$ % "# %

" $ " #&

$(%%#&*)*'!

5. [ (non-straight)]T 0 6?=2 !!"*((("%#

% "$" &

&

% &"!)#&*)*'! #&*)*'!

0

6. [ (non-flush)] 0 2 6468T => ! $* %

"! %" "

&

5

% "!# !#&*)*'! #&*)*'!

0 0

7. [ - - - ]T >2


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Summary Table of Poker Hand ProbabilitiesPoker Hand Number of Hands (without regard to card order) Probability

royal (straight) flush 4 .000001539177169

straight (non-royal) flush 36 .00001385169452

four-of-a-kind 624

full house 3,744

flush (non-straight) 5,108

straight (non-

!!!#%!!*'!$)%

!!"%%!&('#!!"*((("%#

flush) 10,200 0 2 6468

three-of-a-kind 54,912

two pairs 123,552

one pair 1,098,240

nothing 1,302,540

! $* %

!#""#)%&"%

!%(&$*!"&'

%##&'*!#('

&!""(($*%

Totals 2,598,960 1.00000000

19. (A negative binomial probability). Suppose your probability of making a free-throw is .75 and the outcome of each free-throw is independent of the other free-throws. The coach makes you attempt free-throws until you have a made a total of30 free throws. Let denote the total number of attempts. Find andT R 8 $IR .

T R 8 (& #& 8 $! $" $!8"

#*$! 8$! for

and

IR (&IR lW #&IR lJ$! $! $!

(&" IR #&" IR #* $!

(&IR (&IR "$! #*

IR IR $! #*%$

... 30( ) IR %!$!43


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20. I have 3 children Adam, Barbara, and Chris. I own 10 identical Rolex watches. When I die I shall bequeath the watches to my children. If every possibledistribution of the watches is equally-likely, what is the probability that Adam gets 2watches, Barbara gets 8 watches, and Chris gets none?

T E # F ) G ! " "'' "##

.


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THE

END

CLPRPBKY Classical Probability

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