Question

3 people think of a number from {1,2,3,4,5,6,7}
show that the probability that
a) the 3 integers are different, given that the least integer selected is 5 equates to 6/19

b) the sum of the 3 integers is more than 15, equates to 56/343

** I have reworded the question as the previous answers were not correct

Answer 1

I get different answers:
a) 1/7
b) 5/21

Answer 2

one approach that i have taken is:
1. given that one person has selected 5 then picking the 6 is 1/6 and picking the 7 is 1/7 so that equates to 1/49.

2. however there is no stipulation on the order so the there are 3! ways to pick the 5,6,7
so we get 6/49.

Answer 3

could the 6/19 be a typo?

Answer 4

i dunno, i assumed order didn't matter

Answer 5

up till now my text has been very accurate, but that question has me stumped

Answer 6

any idea about part b

Answer 7

there are 63 total combinations
7C1 + 7C2 + 7C3 = 7 + 21 + 35 = 63

Answer 8

for part b) i attempted it like so:
all unique numbers summing > 15 , max 7
7+6+5, 7+6+4, etc
and progressed through those, each choice can be arranged 3! ways
and then went for double entries like
7+7+5 etc that can be arrange 3 ways
added then up
I think that works but it took a long time
there has to be a better way!!!

Answer 9

Only 15 of which have sum greater than 15:
666
777
367
467
567
556
557
664
665
667
772
773
774
775
775

Answer 10

missing 754

Answer 11

some of them can be chosen in multiple ways making them more likely
so 367,467,754,567 can each be selected in 3! or 6 ways giving 24

Answer 12

oh yeah, so its 16/63 then
..i thought order didn't mat

Answer 13

there are 5 with 2 of the same digits which can be selected each in 3 ways giving another 15

Answer 14

sorry 7 with 2 digits
so thats 21

Answer 15

hey thats 56/343
i count 10 with 2 digits...thats 30
plus 24
plus 2

Answer 16

plus the 2 with unique numbers totals 24+21+2 out of 7*7*7 gives us 47/343

Answer 17

hmmm i think we must be missing some

Answer 18

765
764
763
754
each avail in 6 combinations = 24
776
775
774
773
772
766
755
655
665
664
each avail in 3 combinations = 30
666
777
each availin 1 combination = 2

out of 7*7*7 = 343

equals 56/343

Answer 19

However there has to be a much better way than that, it was a past paper question and that took ages!

Answer 20

Please correct my justification for part a)

The total ways of getting 5,6,7 in any order is 6

The total ways of selecting any three numbers (inc. repetitions) is 7^3

So why not 6/343

Answer 21

the question says that 'given that 5 is the least selected number' - could that be it? I find the terminology is confusing !

Answer 22

I think that means that 5 is given as selected
I also guess that 5 doesn't have to be the first selected?

Answer 23

so we are really saying what is prop of selecting 6 and a 7 out of the selection, but the 5 could be in any order

Answer 24

** that is how i read it