Math of Bonus Heroes, worst case? (aka: check my math)

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  • Quebbster
    Quebbster Posts: 8,070 Chairperson of the Boards
    deadtaco wrote:
    Quebbster wrote:
    If I were in your shoes, I'd probably just wait until I got a red Star-Lord cover through other means (a progression reward, most likely)

    The issue there is that covers from progression (or reward) tend to be of only one color variety. They may have to wait until Starlord has been the 4* reward 3 times before getting the red cover. With 45 (or whatever) 4*s now that would take ages.
    Yes, but if Starlord is not a favorite hero there is no chance yellow or purple covers will drop from tokens, so time is not necessarily an issue.
  • MissChinch
    MissChinch Posts: 509 Critical Contributor
    Sorry OP, got sidetracked from your original question about the worst case... so instead of **** on the thread some more, straight to the point about the math, I think you're close, but this is how I calculate it out:

    The odds stated in game for a legendary token to be a 5* is 1 in 7, leaving 6 in 7 for a 4*. The fractions of percentages make a difference, so I'll just carry them through since I don't know where the game rounds.

    Odds of getting a bonus 4* is 5% of the 6/7 chance you have of pulling a 4*

    Assuming cover distribution of each color cover is equal then you're at 1/3rd of that being red.

    So, multiplying that all out: (1 * 5 * 6)/(3 * 100 * 7) = (2*3*5)/(2*2*3*5*5*7) = 1/(2*5*7)
    = 1/70

    This is just the odds of pulling... your worst case is simply that you never pull the cover you want, its entirely possible you never pull a single bonus hero let alone one of the right color, so the best we can do is calculate your odds of having at least 3 reds over a series of pulls..

    Its far easier to count the condition you don't want to have happen, thats pulling no covers you're interested in, pulling just one cover you're interested in, and only pulling two covers you're interested in... 3 covers is good, but scenarios where you pull 4, 5 or 6+ covers are also good, so its easier for me to count what you don't want to have happen and subtract those odds from 1.

    chance of success =
    1 -
    (chance of pulling no covers) -
    (chance of pulling just one cover) -
    (chance of pulling just two covers)

    chance of pulling no covers = (1-1/70)^N (N being the number of pulls)
    chance of pulling just one cover = (1/70)(1-1/70)^(N-1) (N being the number of pulls)
    chance of pulling just two covers = (1/70)^2(1-1/70)^(N-2) (N being the number of pulls, N>=2)

    putting it all together:
    Chance of success over N pulls =
    1 - (1-1/70)^N - (1/70)(1-1/70)^(N-1) - (1/70)^2(1-1/70)^(N-2)

    You can put that in excel or something to toy around with if you're interested, the results might just be depressing...
    You hit 50% likely to be satisfied with just 50 pulls...
    75% chance to be successful after 98 pulls...
    90% chance to be successful after 210 pulls...
    99% chance to be successful after 322 pulls...

    Hope that's useful, or at least interesting...
  • Dayv
    Dayv Posts: 4,449 Chairperson of the Boards
    Straycat wrote:
    The whole point of this thread was about covering one specific character at 5/5/0, not about vaulting in general.
    Sorry, but right now, every thread on the forum is about vaulting. icon_e_wink.gif
  • Dayv
    Dayv Posts: 4,449 Chairperson of the Boards
    My suggestion is to avoid this math completely by never selecting someone as a bonus hero who has a color you can't immediately use. Either set it to someone you're trying to cover who is at 4 covers or less on all three powers, or only set them to champs. I'm doing the only-champs route.

    Yeah, it sucks that this feature doesn't get around the problems of unusable covers, but this is a rare case where you can slow down the rate of unusable rewards just a little by making decisions that don't rely as much on hope and luck.
  • ClydeFrog76
    ClydeFrog76 Posts: 1,350 Chairperson of the Boards
    I've barely even seen gold from my pulls since it got implemented, let alone BHs.
  • smkspy
    smkspy Posts: 2,024 Chairperson of the Boards
    ClydeFrog76 wrote:
    I've barely even seen gold from my pulls since it got implemented, let alone BHs.

    Yeah, I noticed that too, especially elites. Been hitting a hard streak of 2 stars ever since BH started.
  • j0nats
    j0nats Posts: 149 Tile Toppler
    I wouldnt waste my 5% bonus chance in this case cos theres a 2/3 probability the cover would be purple or yellow.

    Also starlords best powers are his yellow n purple anyway.. furthermore he's great underlevelled so he can hide behind his teammates
  • herm1978
    herm1978 Posts: 153 Tile Toppler
    j0nats wrote:
    I wouldnt waste my 5% bonus chance in this case cos theres a 2/3 probability the cover would be purple or yellow.

    Also starlords best powers are his yellow n purple anyway.. furthermore he's great underlevelled so he can hide behind his teammates

    Just not much use in his crash icon_e_sad.gif

    (and no, I'm not going to make him a "favourite" until they fix the duplicate cover problem)
  • smoq84
    smoq84 Posts: 421 Mover and Shaker
    I learned yesterday that math doesn't apply to this game... I've spend 900cps for 36 latest tokens and got:
    0x 5*'s
    0x bonus 4* covers

    :/
  • broll
    broll Posts: 4,732 Chairperson of the Boards
    DayvBang wrote:
    My suggestion is to avoid this math completely by never selecting someone as a bonus hero who has a color you can't immediately use. Either set it to someone you're trying to cover who is at 4 covers or less on all three powers, or only set them to champs. I'm doing the only-champs route.

    Yeah, it sucks that this feature doesn't get around the problems of unusable covers, but this is a rare case where you can slow down the rate of unusable rewards just a little by making decisions that don't rely as much on hope and luck.

    That's pretty much my take on this as well. Which sucks when my Hulkbuster is 5/3/3, Teen Jean is 5/2/3, Thoress is 5/5/0, Iceman is 0/5/4, Nova is 5/2/2, etc. etc...
  • Chrono_Tata
    Chrono_Tata Posts: 719 Critical Contributor
    MissChinch wrote:
    chance of pulling no covers = (1-1/70)^N (N being the number of pulls)
    chance of pulling just one cover = (1/70)(1-1/70)^(N-1) (N being the number of pulls)
    chance of pulling just two covers = (1/70)^2(1-1/70)^(N-2) (N being the number of pulls, N>=2)

    putting it all together:
    Chance of success over N pulls =
    1 - (1-1/70)^N - (1/70)(1-1/70)^(N-1) - (1/70)^2(1-1/70)^(N-2)
    Just a small correction: you also need to multiply the number of combinations that you can pull 1 and 2 covers into their respective probabilities as well, as the formula (P)^(n)*(1-P)^(N-n) only calculates the probability of a single combination. For example, if you draw 4 times, the possible combinations of getting exactly 1 success is 4, the possible combinations of getting exactly 2 success is 6, etc. You can use the =COMBIN() function in excel to calculate this.

    So the correct formula to calculate the success will be:
    1- (1-P)^N- P*(1-P)^(N-1)*COMBIN(N,1) - P^(2)*(1-P)^(N-2)*COMBIN(N,2)

    (First COMBIN left out as COMBIN(N,0) is always equal to 1)
  • MissChinch
    MissChinch Posts: 509 Critical Contributor
    Chrono_Tata wrote:

    Just a small correction: you also need to multiply the number of combinations that you can pull 1 and 2 covers into their respective probabilities as well, as the formula (P)^(n)*(1-P)^(N-n) only calculates the probability of a single combination. For example, if you draw 4 times, the possible combinations of getting exactly 1 success is 4, the possible combinations of getting exactly 2 success is 6, etc. You can use the =COMBIN() function in excel to calculate this.

    So the correct formula to calculate the success will be:
    1- (1-P)^N- P*(1-P)^(N-1)*COMBIN(N,1) - P^(2)*(1-P)^(N-2)*COMBIN(N,2)

    (First COMBIN left out as COMBIN(N,0) is always equal to 1)

    You are absolutely correct, and that's a little embarrassing, especially after I went through writing it all out, that's what I get for thinking I remember stuff instead of googling it...

    Sorry OP, there are N ways to choose a single cover and N choose 2 ways to choose two covers over a series of N pulls that I completely left out, just like Chrono_Tata says...

    That makes the probabilities of the cases you DONT like higher, which increases how bad it sucks...
  • GurlBYE
    GurlBYE Posts: 1,218 Chairperson of the Boards
    smoq84 wrote:
    I learned yesterday that math doesn't apply to this game... I've spend 900cps for 36 latest tokens and got:
    0x 5*'s
    0x bonus 4* covers

    :/


    The issue is too many people on this forum seem to use this math under the assumption hat every draw is a vault in their math, or that 5% means 1 in 20 on average of 20 should be there.


    Each individual draw is the percentage you seek.
    It resets every time. There is no mystery counter that the fates themselves spin to make sure you get your percentage.

    It's possible some people are getting 5 bonus draws a day while some people will get 1 a month.

    All calculating the percentages over and over sorta seems to do is set expectations that we've seen won't be met.

    Like the advertised percentages on 3's on elites is the "same" as far as tokens say, but I've exclusively gotten 2's from elites since bonus heros started. Compared to when they released, when I opened my first 9 and 8 were 3's.

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