Idolm@ster SS: 10-Roll compared with 41 daily rolls

Cost of daily single roll =60 Gems
Cost of 10 roll = 2500 Gems
-> 2500 gems worth of daily rolls -> ~41 rolls

Chance of getting an SR -> 10%
Chance of getting an SSR-> 1,5%

Chance of getting neither -> 88,5%

Thus, if we use the n over k formula for bernoulli-processes we get the following results:

[41 daily rolls]
Chance of getting exactly x amount of SR/SSR cards
0= 0,0067
1= 0,0356
2= 0,0925
3= 0,1562
4= 0,1928
5= 0,1854
6= 0,1446
7= 0,0939
8= 0,0519
9= 0,0247
10=0,0103

As those add up to ~0,95, the chance for getting 11-40 SR/SSR is 5%. I'll leave those out, as those would be too much, and I think this much data is already enough to conclude which roll is superior (I'm writing this after having calculated the 10 roll already)

Some conclusions we can draw from this
-> With a chance of 99,6% you get at least one SR/SSR (One might argue that counts as basically "guaranteed")
-> With a chance of over 50% you get 3-6 SR/SSR
-> The expected average value is 4,715 SR


Let's compare this to the 10-Roll

One card is a guaranteed 100% SR. Meaning we effectively work with 9 instead of 10 rolls for this calculation and add +1 to our x afterwards
[10 Roll]
Chance of getting exactly x amount of SR/SSR cards
0= 0,3330
1= 0,3894
2= 0,2025
3= 0,0614
4= 0,012
5= 0,0015

We now add the guaranteed SR

1= 0,3330
2= 0,3894
3= 0,2025
4= 0,0614
5= 0,012
6= 0,0015
They add up to ~0,998 -> the chance of getting 7 and above is 0,002

The expected average value of SR/SSR is 2,035

I thus conclude that 41 daily rolls are worth it far more, than 10 rolls. Daily rolls are not a gamble (in relative terms at least, compared to the other option), they are in fact the smarter choice of the two, if you wish to sink money into the game.