Prize Probabilities

Setup

library(extraDistr)

Ignoring basics:

(Note: this is overly complex and could have been done with just a single dhyper, but I didn’t realize at the time and wanted to reuse this code for the more complex cases)

handProbs <- data.frame(matrix(0, nrow=59, ncol=8)) # stores the probability that a specific number of copies of the target card are in the starting hand
prizeProbs <- data.frame(matrix(0, nrow=59, ncol=8)) # stores the probability that the a specific number of target cards are in the prize cards
colnames(prizeProbs) <- paste(0:6, "Prized")
colnames(prizeProbs)[8] <- "Expected Number Prized" 

row.names(prizeProbs) <- paste(1:59, "Copy(s) of Card Total")

colnames(handProbs) <- paste(0:7, "In Hand")
row.names(handProbs) <- paste(1:59, "Copy(s) of Card Total")

for(n in 1:59){ # n is the number of copies of the target card in deck, which must be between 1 and 59
  indivProb <- dhyper(0:(min(n,7)),n,60-n,7) # probabilities that 0 to 7 of the target card are in the starting hand
  for(i in max(n-52,1):min(length(indivProb),8)){ # for every non zero probability of copies of target in hand
    curHandProb <- as.numeric(indivProb[i]) # probability that i copies of the target card are in the starting hand
    handProbs[n,i] <- curHandProb
    curPrizeProb <- dhyper(0:min(n,6),n-i+1,52+i-n,6) # probabilities that 0 to 6 of the target card are in the prize cards (i is 1 when 0 are prized, so it needs to be adjusted by 1)
    for(j in 1:length(curPrizeProb)){ # check every probability that 0 to 6 of the target are prized
      prizeProbs[n,j]<-prizeProbs[n,j]+curHandProb*curPrizeProb[j] # add the probability that that number of target cards are prized AND i target cards are in hand to the probability of having j prized
    }
  }
  for(i in 1:6){
    prizeProbs[n,8] <- prizeProbs[n,8] + prizeProbs[n, (i+1)]*i # calculate expected value of number prized
  }
  
}
write.csv(prizeProbs,file="output/ignoreBasics/prizeProbabilitiesNoBasics.csv")
write.csv(handProbs,file="output/ignoreBasics/handProbabilitiesNoBasics.csv")

The Efficient Method

prizeProbs <- data.frame(matrix(0, nrow=59, ncol=7)) # stores the probability that the a specific number of target cards are in the prize cards
colnames(prizeProbs) <- paste(0:6, "Prized")

row.names(prizeProbs) <- paste(1:59, "Copy(s) of Card Total")

for(n in 1:59){ # n is the number of copies of the target card in deck, which must be between 1 and 59
  prizeProbs[n, ] <- dhyper(0:6,n,60-n,6) # probabilities that 0 to 6 of the target card are in the prize cards
}

Taking basics into account:

basicsInHandProbs <- matrix(0, nrow=59, ncol=9)
basicsInHandProbs <- data.frame(basicsInHandProbs)
colnames(basicsInHandProbs) <- paste(1:7, "Basic(s) In Hand")
colnames(basicsInHandProbs)[8] <-"Expected number of hands seen"
colnames(basicsInHandProbs)[9] <-"Expected number of mulligans"
row.names(basicsInHandProbs) <- paste(1:59, "Basic(s) In Deck")
mat <- matrix(0, nrow=28, ncol=3) # needed for the multivariate hypergeometric distribution 
count <- 0
for(b in 1:7){
  for(t in 0:(7-b)){
    count<- count+1
    mat[count,] <- c(7-b-t, b, t) # adds every combination of basic, target, and other cards that adds up to 7 and has at least one basic to the matrix of quantiles for the distribution later
  }
}
for(b in 1:59){ # b is the number of basics in deck
  prizeProbs <- data.frame(matrix(0, nrow=60-b, ncol=7)) # needs to be remade for every number of basics
  handProbs <- data.frame(matrix(0, nrow=60-b, ncol=8)) # stores the probability that a specific number of copies of the target card are in the starting hand
  
  colnames(prizeProbs) <- paste(0:6, "Prized")
  row.names(prizeProbs) <- paste(1:(60-b), "Copy(s) of Card Total")
  
  colnames(handProbs) <- paste(0:7, "In Hand")
  row.names(handProbs) <- paste(1:(60-b), "Copy(s) of Card Total")
  for(n in 1:(60-b)){ # n is the number of copies of the target card in the deck
    curHandSuccessProb<-1-dhyper(0,b,60-b,7)
    handCombos<-dmvhyper(mat, c(60-b-n, b, n), 7)/(curHandSuccessProb) # p(having 1...7 basics and 0...6 targets in starting hand)/(1-p(not having basic in starting hand))
    basicsInHandProbs[b, 1] <- sum(handCombos[1:7])
    basicsInHandProbs[b, 2] <- sum(handCombos[8:13])
    basicsInHandProbs[b, 3] <- sum(handCombos[14:18])
    basicsInHandProbs[b, 4] <- sum(handCombos[19:22])
    basicsInHandProbs[b, 5] <- sum(handCombos[23:25])
    basicsInHandProbs[b, 6] <- sum(handCombos[26:27])
    basicsInHandProbs[b, 7] <- sum(handCombos[28])
    expectedHands <- 1/(curHandSuccessProb) # expected value of a geometric distribution is 1/p, so 1/(1-P(mulligan))
    basicsInHandProbs[b, 8]<- expectedHands
    basicsInHandProbs[b, 9]<- expectedHands-1 # the starting hand itself is one hand, so expected number of mulligans is one fewer than the expected number of hands seen
    for(c in 1:length(mat[,1])){
      if(handCombos[c]!=0){
        h <- mat[c,3] # number of the target card in hand
        handProbs[n,h+1]<- handProbs[n,h+1]+handCombos[c] # adds to the probability of having h copies of the card in the starting hand
        curPrizeProb <- dhyper(0:6,max(0,n-h),min(53+h-n, 53),6) # actual probabilities for having 0...6 copies of the card prized given that h copies are in hand and n total
        
        for(j in 1:length(curPrizeProb)){ 
          prizeProbs[n,j]<-prizeProbs[n,j]+curPrizeProb[j]*handCombos[c] # add the probability of having j copies prized given that we got this hand arrangement AND getting the hand arrangement of mat[c,] to the total probability of having j copies prized
        }
        
      }
    }
    
  }
  write.csv(handProbs,file=paste0("output/notBasic/handProbabilities_", b,"Basics.csv"))
  write.csv(prizeProbs,file=paste0("output/notBasic/prizeProbabilities_", b,"Basics.csv"))
}
write.csv(basicsInHandProbs,file=paste0("output/basicsInHand.csv"))

If target is a basic:

mat <- matrix(0, nrow=36, ncol=3) # needed for the multivariate hypergeometric distribution 
count <- 0
for(b in 0:7){ # since the target card is a basic now, we can also have 0 basics but one target
  for(t in max(0, 1-b):(7-b)){
    count<- count+1
    mat[count,] <- c(7-b-t, b, t) 
  }
}
for(b in 0:59){ # b is the number of other basics in deck
  prizeProbs <- data.frame(matrix(0, nrow=min(4,60-b), ncol=7)) # needs to be remade for every number of basics
  handProbs <- data.frame(matrix(0, nrow=min(4,60-b), ncol=8)) # stores the probability that a specific number of copies of the target card are in the starting hand
  
  colnames(prizeProbs) <- paste(0:6, "Prized")
  row.names(prizeProbs) <- paste(1:min(4,60-b), "Copy(s) of Card Total")
  
  colnames(handProbs) <- paste(0:7, "In Hand")
  row.names(handProbs) <- paste(1:min(4,60-b), "Copy(s) of Card Total")
  for(n in 1:min(4,60-b)){ # n is the number of copies of the target card in the deck
    curHandSuccessProb<-1-dhyper(0,b+n,60-b-n,7)
    handCombos<-dmvhyper(mat, c(60-b-n, b, n), 7)/(curHandSuccessProb) # p(having 1...7 basics and 0...7 targets in starting hand)/(1-p(not having basic in starting hand))
    
    for(c in 1:length(mat[,1])){
      if(handCombos[c]!=0){
        h <- mat[c,3] # number of the target card in hand
        handProbs[n,h+1]<- handProbs[n,h+1]+handCombos[c] # adds to the probability of having h copies of the card in the starting hand
        curPrizeProb <- dhyper(0:6,max(0,n-h),min(53+h-n, 53),6) # actual probabilities for having 0...6 copies of the card prized given that h copies are in hand and there are n total
        
        for(j in 1:length(curPrizeProb)){ 
          prizeProbs[n,j]<-prizeProbs[n,j]+curPrizeProb[j]*handCombos[c] # add the probability of having j copies prized given that we got this hand arrangement AND getting the hand arrangement of mat[c,] to the total probability of having j copies prized
        }
        
      }
    }
    
  }
  write.csv(handProbs,file=paste0("output/isBasic/handProbabilitiesB_", b,"Basics.csv"))
  write.csv(prizeProbs,file=paste0("output/isBasic/prizeProbabilitiesB_", b,"Basics.csv"))
}