Question

A die is thrown four times. What is the probability that each number thrown is at least as high as all the numbers that were thrown earlier?

Answer 1

hard

Answer 2

If the first three dice are 1, 1, 1 then the last one can be anything so here we have 6 success cases.
If they are 1,1,2 then only 5 possibilities are possible, as the last one has to be 2 or more
If 1,1,3, then 4 possibilities, if 1,1,4,then 3 possibilities, if 1,1,5, then 2 possibilities and if 1,1,6 then 1 possibility only so if the first two dice are 1,1 we have 6+5+4+3+2+1=21 possibilities.
Clearly, now, if the first two dice are 1,2 then it is the same as above starting at the 1,1,2 case (which is now 1,2,2) so the possibilities are 5+4+3+2+1=15
If 1,3 then 4+3+2+1=10
If 1,4 then 3+2+1=6,
If 1, 5 then only 2+1=3 and if 1,6 then only one possibility
So, if the first die is 1 the possibilities are 21+15+10+6+3+1=56
Therefore, if the first die is 2 the possibilities are 15+10+6+3+1=35
If the first die is 3 the possibilities are 10+6+3+1=20
If the first die is 4 the possibilities are 6+3+1=10
If the first die is 5 the possibilities are 3+1=4 and
if the first die is 6 there is only 1 possibility
So the total number of possibilities of success are 56+35+20+10+4+1 = 126
The total number of cases is 6^4=1296
Therefore the probability is 126/1296 = 7/72 = 0.097222

Answer 3

wow nice job
grind it till you find it!

Answer 4

the # of favorable outcomes can be obtained using the "stars and bars" approach.
consider distinct ordered boxes 1-2-3-4-5-6 in which 4 identical balls are to be placed.
this can be done in (4+6-1)C(6-1) = 9C5 = 126 ways
[for how this formula comes about, see link ]
http://jhyun95.hubpages.com/hub/Stars-and-Bars-Combinatorics

total ways to roll 4 dice = 6^4 = 1296

Pr = 126/1296

Answer 5

I'm trying to find the correlation between your solution and the procedures I was taught in class. I'm still a little confused, but I will definitely take this answer into consideration and report back here after my professor goes over it. Thank you for your help. :)

Answer 6

If all else fails, use logarithms/complex numbers.