One of the toughest question for statisticians!!?

a) If there are numbers between 1 and 40, and 7 are to be drawn at random, what are the possible number of outcomes?


b) Imagine if I write 1,2,3,4,5,6 and 7 on a note book, and 1 - 40 written on small pieces (i.e. 40 small pieces of paper of paper) kept in a box, what is the probability that the 7 numbers i draw from the box are:


i) the same 7 number


ii) 6 matching numbers


iii) 5 matching numbers


iv) 4 matching numbers


v) 3 matching numbers


vi) 2 matching numbers


vii) 1 matching number


viii) no matching numbers





C) does the number of possible outcomes has an effect on the probability of matching numbers?





P.S. numbers once drawn cannot be repeated





please give formulae aswell

One of the toughest question for statisticians!!?
a)40*39*38*37*36*35*34=40P7 (nPr function) = 93,963,542,400 possible outcomes





b) use hypergeometric distribution


(n,r) is the nCr function


i) (33,0)*(7,7)/(40,7) = 1/18642560


ii)(33,1)*(7,6)/(40,7) =77/6214520


iii)(33,2)*(7,5)/(40,7)=462/776815


iv)(33,3)*(7,4)/(40,7)=4774/466089


following the pattern the rest are:


v)11935/155363


vi)207669/776815


vii) 969122/2330445


viii)178002/776815





c) no, the number of possible outcomes is just a number. the number of selections, in this case 7, is what affects the probability.
Reply:If you're asking what the probability of winning a 7-number lottery with 40 possibilities for the numbers, here goes:





i) the same 7 number





probability = 1/N





where N = [40c7]


where [40c7] = 40! / (40-7)!*(7!)


where (for example) 7! = 7*6*5*4*3*2*1








ii) 6 matching numbers





probability = M / N





where N = [40c7]


and M = [7c6] * [33c1]








in general, to match "x" numbers


probability = M / N





where N = [40c7]


and M = [7cx] * [40c(7-x)]
Reply:I tend to agree. That is tough.
Reply:for question a):





There are 40 choices for the 1st number, then 39 choices for the 2nd number, 38 choices for the 3rd number, etc.





so the answer is 40 x 39 x 38 x 37 x 36 x 35 x 34 =


93 963 542 400.





for question b)





Since you specified that numbers oncedrawn cannot be repeated, then there will be no matching numbers. The answer is zero for i-vii. For vii, the answer is 1 (ie you are sure that there are no matching numbers)





for question c)





No, the event of having matcing numbers is indpendent of having an outcome.
Reply:a)





are you talking about how many combinations or permutations of 7 numbers drawn from 40?





Permutations, order matters. in general if you have n objects and you select k of them, you have n! / (n - k)! permutations.





Combinations, order does not matter, in general if you have n objects and you select k of the you have n! / ( k! * (n - k)! ) combinations.





Here if you're talking about permutations then you have 40! / (40 - 7)! = 93963542400





Here if you're talking about combinations the you have 40! / ( 7! * (40 - 7)! ) = 18643560





-----------------------





to match the same seven numbers you have 1 / 18643560 chance of matching all seven





here is a short hand for combinations 40nCr7 means 40 choose 7.





ii. 7nCr6 * (1 / 40nCr6) = 1 / 548340


iii. 7nCr5 * (1 / 40nCr5) = 7 / 219336


iv. 7nCr4 * (1 / 40nCr4) = 7 / 18278


v. 7nCr3 * (1 / 40nCr3) = 7 / 1976


vi. 7nCr2 * (1 / 40nCr2) = 7 / 260


vii. 7nCr1 * (1 / 40nCr1) = 7 / 40


viii. 7nCr0 * (1 / 40nCr0) = 1 - (sum of the seven above) = 1 - 7 / 34 = 27 / 34





----------





the number of outcomes plays a big part in probabilities. in general a probability can be defined as the number of outcomes you are interested in divided by all possible outcomes.
Reply:part a.


There is a possible


40 x 39 x 38 x 37 x 36 x 35 x 34 = 93,963,542,400 outcome





part b.


7 number the same


7/40 x 6/39 x 5/38 x 4/37 x 3/36 x 2/35 x 1/34 = 5040/93,963,542,400 or 0.00000536378%





6 number the same


7/40 x 6/39 x 5/38 x 4/37 x 3/36 x 2/35 = 5040/2,763,633,600 or 0.0001823686%





5 number the same


7/40 x 6/39 x 5/38 x 4/37 x 3/36 = 2520/78,960,960 or 0.00319145%





4 number the same


7/40 x 6/39 x 5/38 x 4/37 = 840/2,193,360 or 0.0382974%





3 number the same


7/40 x 6/39 x 5/38 = 210/59,280 or 0.354251%





2 number the same


7/40 x 6/39 = 42/1560 or 2.6923%





1 mumber the same


7/40 = 17.50%





no number the same


33/40 x 32/39 x 31/38 x 30/37 x 29/36 x 28/35 x 27/34 = 2,153,112,920/93,963,542,400 or 22.9143361%





part c.


No
Reply:that's a lot of questins.





a) 40 * 39 * 38 ... * 34 or 40!/(40-7)!





the magnitude of this number is a little less than 4 * 10^10





b)


i) would be one in 40!/(40-7)!


ii) would be more like one in 40!/(40-6)! - 40!/(40-7)!


iii) would be more like one in 40!/(40-5)! - 40!/(40-6)! - 40!/(40-7)!


etc.

orange