Population Genetics Lab

Lab Instructor: Hari Chhetri PhD student

Department of Biology Area of Research: Association genetics (Populus) Office: Life Sciences Building, Room # 5206. Office Hour: T, W, F – 11:30 AM – 12:30 PM or by appointment. Email ID: harichhetri@gmail.com Tel. #: 304-293-6181

Probability  and  Popula/on  Gene/cs  

 Popula/on  gene/cs  is  a  study  of  probability   Sampling  alleles  from  popula/on  each  genera/on  

A

AA

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A

aa a

a

A

Probability  

Frequen'st  Approach  •  Determine  how  o;en  you  

expect  event  A  to  occur  given  a  LONG  series  of  trials  

Bayesian  Approach  •  Determine  the  plausibility  

of  event  A  given  what  you  already  know  (prior).  

Probability   Measure  of  chance.                P(E)  =    #  of  favorable  outcome  /  Total  #  of  possible  outcome     It  lies  between  0  (impossible  event)  and  1  (certain  event).          Ex.  What  is  the  probability  of  geGng  a  head  in  one    toss  of  a  balanced  coin.  

           Total  possible  outcomes  =  2  (  H,  T)            #    of  Heads  =    1  (H)              P(H)    =    1/  2    =  0.5    =  50  %        

Sample-­‐  point  method  :  1.  Define  sample  space  (S):  Collec/on  of  all  possible  outcomes  of  a  random  expt.    Ex.    S  (Coin  tossed  twice)  

                 2.  Assign  probabili'es  to  all  sample  points    Ex.    P(HH)  =  ¼  ;  P(HT)  =  ¼  ;  P(TH)  =  ¼  ;  P(TT)  =  ¼  

                     

Outcome   1   2   3   4  

First  Toss   H   H   T   T  

Second  Toss   H   T   H   T  

Shorthand   HH   HT   TH   TT  

Sample-­‐  point  method  :  3.Determine  event  of  interest  and  add  their  probabili'es.    Ex.  Find  the  probability  of    geXng  exactly  one  head  in  two  tosses  of  a  balanced    coin.                      

   i.  S  (Coin  tossed  twice)  {  HH,  HT,  TH,  TT}.                                              ii.  P(HT)  =  ¼  ;  P(TH)  =  ¼    

                 iii.  P(HT)  +  P  (TH)  =  ¼  +  ¼    =  2/4  =  ½  .    If  all  sample  points  have  equal  probabili'es  then  –      

       P(A)  =  na  /  N        where,  na  =  #  of  points  cons/tu/ng  event  A  and    N=  Total  #  of  sample  points.              

Sample-­‐  point  method  :    Example:  Use  the  Sample  Point  Method  to  find  the  probability  of  geXng  exactly  two  heads  in  three tosses of  a  balanced  coin.    1.  The  sample  space  of  this  experiment  is:            

                     

2.  Assuming  that  the  coin  is  fair,  each  of  these  8  outcomes  has  a  probability  of  1/8.    3.  The  probability  of  geXng  two  heads  is  the  sum  of  the  probabili/es  of  outcomes  2,  3,  and  4  (HHT,  HTH,  and  THH),  or  1/8  +  1/8  +  1/8  =  3/8  =  0.375.      In  above  example,  find  the  probability  of  geGng    at  least  two  heads.  Solu'on:  1/8  +  1/8  +  1/8+  1/8  =  1/2    

Outcome Toss 1 Toss 2 Toss 3 Shorthand Probabilities 1 Head Head Head HHH 1/8 2 Head Head Tail HHT 1/8 3 Head Tail Head HTH 1/8 4 Tail Head Head THH 1/8 5 Tail Tail Head TTH 1/8 6 Tail Head Tail THT 1/8 7 Head Tail Tail HTT 1/8 8 Tail Tail Tail TTT 1/8

Problem  1:  The  game  of  “craps”  consists  of  rolling  a  pair  of  balanced  dice  (i.e.,  for  each  die  geGng  1,  2,  3,  4,  5,  and  6  all  have  equal  probabili'es)  and  adding  up  the  resul'ng  numbers.  A  roll  of  “2”  is  commonly  called  “snake  eyes”  and  causes  an  instant  loss  when  rolled  in  the  opening  round.  Using  the  Sample-­‐Point  Method,  find  the  exact  probability  of  a  roll  of  snaked  eyes.  (Time  :  10  minutes)  

Probability  

For  large  sample  space:  Use  fundamental  coun'ng  methods.    1.  mn  rule  :  If  there  are  “m”  elements  from  one  group  and  “n”  elements  from  another  group,  then  we  can  have  “mn”  possible  pairs,  with  one  element  from  each  group.  

           

       

   

   mn=  6*6=  36  .      

Second die 1 2 3 4 5 6

Firs

t die

1 2 3 4 5 6

For  large  sample  space  :  Use  fundamental  coun'ng  methods.    2.  Permuta'on:    Ordered  set  of  “r”  elements,  chosen  without  replacement,  from  “n”  available  elements.              Remember:      0!  =  1  (By  defini/on)  

     n!  =  n*(n-­‐1)*(n-­‐2)*…………*2*1.    Example:    How  many  trinucleo'de  sequences  can  be  formed  without  repea'ng  a  nucleo'de  ,  where  ATC  is  different  from  CAT?    Solu'on:  n  =  4  (  A,  T,  C  and  G)                      r  =  3  

       

               =    24.    

)!(!rnnPnr −

=

)!34(!4−

=nrP

For  large  sample  space  :  Use  fundamental  coun'ng  principle.    3.  Combina'on:    Unordered  set  of  “r”  elements,  chosen  without  replacement,  from  “n”  available  elements.              Example:    How  many  trinucleo/de  sequences,  can  be  formed  without  repea/ng  a  nucleo/de  ,  where  ATC  is  same  as  CAT.    Solu'on:  n  =  4  (  A,  T,  C  and  G)              r  =  3  

       

                               =    4.  

 

)!(!!rnr

nC nr −=

)!34(!3!4−

=nrC

For  large  sample  space  :  Use  fundamental  coun'ng  principle.    Problem 2: There are 36 computer workstations in this lab. If there are 18 students in the class, how many distinct ways could students be arranged, with one student per workstation? ( 10 minutes). Problem 3: A local fraternity is organizing a raffle in which 30 tickets are to be sold - one per customer. (10 minutes). a. What is the total number of distinct ways in which winners can be chosen if prizes are awarded as follows: b. If holders of the first four tickets drawn each receive a $30 prize?  

Order  of  Drawing   Prize  

First   $100  

Second   $50  

Third   $25  

Fourth   10$  

Laws  of  Probability  1. Additive law of probability:

B)P(A - P(B) P(A) B)(A P ∩+=∪

P(B) P(A) B)P(A then 0, B)P(A events, exclusivemutually For usly.simaltaneo B andA event of occurrence ofy ProbabilitB)P(A

B.or A event of occurrence ofy ProbabilitB)P(AWhere,

+=∪=∩

=∩

=∪

B)(A Pfor diagramVenn ∪ B)(A Pfor diagramVenn ∩

A B

A and B are Mutually Exclusive

Laws  of  Probability  1. Additive law of probability:

Example: From a pack of 52 cards, one card is drawn at random. Find the probability that the card is “Heart” or “Ace”. Four suits are : Spades, Diamonds, Clubs and Hearts. Each suit has 13 cards: Ace,2,3,4,5,6,7,8,9,10,Jack, Queen and King. There are four of each type, like 4 Aces,4 Jacks, 4 Queens, 4 Kings etc. Solution:

5216

521

524

5213 A)P(H

521 A)P(H ;

524 P(A) ;

5213 P(H)

A)P(H - P(A) P(H) A)(H P

=−+=∪

=∩==

∩+=∪

Laws  of  Probability  2. Multiplicative law of probability:

P(B) P(A) B)(A P ×=∩ (If A and B are independent events)

B)|P(A P(B) A)|P(B P(A) B)(A P ×=×=∩(If A and B are dependent events)

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Example: A pond consists of 50 salmon and 25 trout. Two fish are drawn one by one. Find the probability that both fish are Salmon. a.)with replacement and b.)without replacement

11149

22298

7449

7550

A)|P(B P(A) B)P(A :treplacemen Without :(b) Case

94

7550

7550

P(B) P(A) B)P(A :treplacemen With :(a) Case

Salmon. isdrawn fish second that Prob. : P(B) Salmon. isdrawn fish first that Prob. :P(A) Let,

:Solution

==×=

×=∩

=

×=

×=∩

Problem  4.  An  inexperienced  spelunker  is  preparing  for  the  explora'on  of  a  big  cave  in  a  rural  area  of  Mexico.    He  is  planning  to  use  two  independent  light  sources  and  from  reading  their  technical  specifica'ons,  he  has  concluded  that  each  source  is  expected  to  malfunc'on  with  probability  of  0.01.  What  is  the  probability  that:    a)  At  least  one  of  his  light  sources  malfunc'ons?  b)  Neither  of  his  light  sources  malfunc'on?  (Time  :  15  minutes)        

Problem  5.  GRADUATE  STUDENTS  ONLY:  In  street  craps,  the  opening  toss  wins  if  a  7  or  11  is  rolled,  and  the  “pass”  bets  will  pay  off.  Meanwhile  if  2,  3,  or  12  is  rolled,  only  “don’t  pass”  bets  will  win.      a)  Is  it  safest  to  bet  “pass”  or  “don’t  pass”  on  the  opening  roll?  Show  the  exact  probability  of  each  outcome.    b)  If  the  shooter  rolls  7  three  'mes  in  a  row,  is  it  safest  to  bet  “pass”  or  “don’t  pass”  on  the  next  roll?  Defend  your  answer.      

(Time:  5  minutes)