How to calculate the probability in lottery with returning number?

Question

in lottery type game, 6 number drown, the number can be from
000000-999999 (number can be with returning )
first prize is all 6 numb correct
the odds is 10^6=1000000 for first prize
secondary prize table is:
5 correct number.
4 correct number.
3 "
2 "

how can i calculate secondary prize odds?

Answer 1

How do you get exactly five numbers correct? Let's say it is the last five correct and the first one incorrect, we can write this as *ABCDE where * represents a "wildcard," a non-correct digit while the other positions A,B,C,D,E are correct. There are 9 of these (where * could be any of the 9 wrong digits). Similarly, 9 possiblities for A*BCDE, AB*CDE, ABC*DE, ABCD*E, and ABCDE*, so a total of 6*9=54 possibilties for 5 correct. The probability is 54/1000000.

For four correct, you could write out all the ABCD** ... A*BC*D ... possibilities but there is an easier way, using the binomial coefficient. You count how many to pick where your correct digits are using 6Cn and then the incorrect digits could have any one of 9 possibilities. So the probilities are

4 digits 6C4*9^2/ 1000000 =1215/ 1000000
3 digits 6C3*9^3/ 1000000 = 14580/ 1000000
2 digits 6C2*9^4/ 1000000 = 98415/ 1000000

Answer 2

You did not mention getting 5 out of 6 correct numbers. Therefore the draw is 5 numbers and the order does not matther. the anwer is the chances of getting 5 numbers right x number of way to get it right (e.g the winning number is 1,2,3,4, 5. How many ways you can arrange it. If my ticket is 2,3,5,1,4 I still win, so is 1,5,3,2,4) Your first anwerer gave you the chances of getting 5 numbers right. You should figure out how many ways to arrange 5 non-repeating numbers; someone has answered that part too. An alternative way to set it up would be prob(hitting 1 of the 5 winning numbers from 50) x prob(hitting 1 of the remaining 4 winning numbers from 49) x ..... prob(hitting 1 of the last 1 number from 46). 5/50 x 4/49 x 3/48 x 2/47 x 1/46. This way you don't have look at the permutation part. (also this set up works well if the question is draw 6, and you need to hit 5 numbers..) Hope you enjoy your math class. I did.

Answer 3

To find the required probability, of having 5 numbers out of 6 correct,
Simply take

10x10x10x10x10x 6C5 = 600000