Let us denote by S the set of 6 integers (their inclusive range being 1 to 37) drawn at random in a weekly lottery.
Given S_{ord}= (m_{1}, m_{2} ,m_{3} ,m_{4}, m_{5}, m_{6} ) , where the said integers are placed in increasing order, please answer the following questions:
a) What is the expected value of m_{1}?
b) What is the expected value of m_{6}?
c) If your "guess set" consists of 6 distinct integers, chosen randomly within the defined range, what is the expected quantity of numbers in this set that match the drawing's results?
Please resolve analytically and then compare your answers with the results of a simulation, based on at least 100,000 independent drawings.
For any of the expected values, we need to multiply each value by the probability that that value will occur, then add all the products.
In each probability, the number of total combinations of 37 items chosen 6 at a time (i.e., C(37,6)) will come up as the denominator of the probability fractions.
In the case of the lowest chosen number, m1, the number of ways of a given value being the lowest number is C(37m,5) as the other five numbers need to be drawn from a pool of 37m numbers that are higher than m, and the expected value is Sum{m=1 to 32}(m*C(37m,5)/C(37,6)).
In the case of the highest chosen number, m6, the number of ways of a given value being the highest number is C(m1,5) as the other five numbers need to be drawn from a pool of m1 numbers that are lower than m, and the expected value is Sum{m=6 to 37}(m*C(m1,5)/C(37,6)).
The probability of a given number of matches of numbers between your ticket and the winning drawing set is given by:
C(6,n)*C(376,6n)/C(37,6)
as C(6,n) possibilities exist for which n numbers on your card match a winner, and the remaining drawn 6n numbers must come from a pool of numbers not on your card, and that is 376=31.
When these computations are done, the answers are:
a) 38/7 ~= 5.4285714285714285713
b) 228/7 ~= 32.5714285714285714285 (these two add up to 38)
c) 36/37 ~= 0.9729729729729729729
The computations were done by:
5 All=combi(37,6):Expval=0
10 for M1=1 to 32
20 Expval=Expval+M1*combi(37M1,5)//All
30 next
40 print Expval,Expval/1
105 All=combi(37,6):Expval=0
110 for M6=6 to 37
120 Expval=Expval+M6*combi(M61,5)//All
130 next
140 print Expval,Expval/1
205 All=combi(37,6):Expval=0
210 for Match=0 to 6
220 Expval=Expval+Match*combi(6,Match)*combi(376,6Match)//All
230 print combi(6,Match)*combi(376,6Match)/All:Cu=Cu+combi(6,Match)*combi(376,6Match)/All
240 next
245 print Cu
250 print Expval,Expval/1
A million trials gave these averages in a simulation:
5.430902 32.570788 .972381
using:
DEFDBL AZ
FOR tr = 1 TO 1000000
REDIM used(37): lowest = 99: highest = 0: matches = 0
FOR i = 1 TO 6
DO
n = INT(RND(1) * 37 + 1)
LOOP UNTIL used(n) = 0
used(n) = 1
IF n < lowest THEN lowest = n
IF n > highest THEN highest = n
IF n < 7 THEN matches = matches + 1
NEXT
lowtot = lowtot + lowest: hightot = hightot + highest
matchtot = matchtot + matches
IF tr MOD 1000 = 0 THEN
PRINT lowtot / tr, hightot / tr, matchtot / tr
END IF
NEXT
The matches in the random set were as against the set {1,2,3,4,5,6} as any lotto card set is as good as any other.

Posted by Charlie
on 20130328 19:14:21 