### TWO PAIRS VERSUS THREE OF A KIND ###

# A hand of cards is 'two pairs' if there are two cards of one kind, two cards 
# of a different kind, and a fifth card of a still-different kind. For example, 
# 5H 5S JS JD QS is two pairs.

# A hand of cards is 'three of a kind' if there are three cards of one kind, a 
# fourth card of a different kind, and a fifth card of a still-different kind. 
# For example, KC KS KD 3S 2S is three of a kind.

# Which one is rarer? The answer matters to poker players, because the rarer 
# one beats the more common one.

# We could compute the exact probabilities. In fact, computing them is good 
# practice for you. But we could instead simulate a bunch of randomly chosen 
# hands, and simply count how often we get two pairs or three of a kind. In 
# general, simulation lets us check our exact answers, and it gives us 
# approximate answers, when exact answers are too difficult to compute.

# Let's agree that cards are simply numbers between 1 and 52. The first 13 
# numbers are A, 2, ..., K of spades. Then come the hearts, then the diamonds, 
# then the clubs. A hand is a vector of five distinct cards.

# This function, whenever you invoke it, produces a random hand. (You are not 
# supposed to understand how this code works.)
randomHand <- function() {
  sample(1:52, 5, replace=FALSE)
}

# This function, when you give it a hand, gives you back a sorted list of the 
# kinds in that hand (with repetition). The possible kinds range from 1 (A) to 
# 13 (K).
handKinds <- function(hand) {
  sort((hand - 1) %% 13 + 1)
}

# This function tests whether a given hand's kinds are two pairs.
isTwoPairs <- function(kinds) {
  if (kinds[[1]] == kinds[[3]] || kinds[[3]] == kinds[[5]] || 
      kinds[[5]] == kinds[[1]]) {
    # There are not three distinct kinds in the hand.
    FALSE
  } else if (kinds[[1]] == kinds[[2]] && kinds[[3]] == kinds[[4]]) {
    # The hand looks like pair, pair, other.
    TRUE
  } else if (kinds[[1]] == kinds[[2]] && kinds[[4]] == kinds[[5]]) {
    # The hand looks like pair, other, pair.
    TRUE
  } else if (kinds[[2]] == kinds[[3]] && kinds[[4]] == kinds[[5]]) {
    # The hand looks like other, pair, pair.
    TRUE
  } else
    FALSE
}

# This function tests whether a given hand's kinds are three of a kind.
isThreeOfAKind <- function(kinds) {
  if (kinds[[1]] == kinds[[2]] && kinds[[2]] == kinds[[3]] && 
      kinds[[1]] != kinds[[4]] && kinds[[4]] != kinds[[5]] && 
      kinds[[5]] != kinds[[1]]) {
    # This hand looks like three, other, another.
    TRUE
  } else if (kinds[[2]] == kinds[[3]] && kinds[[3]] == kinds[[4]] && 
             kinds[[1]] != kinds[[2]] && kinds[[2]] != kinds[[5]] && 
             kinds[[5]] != kinds[[1]]) {
    # This hand looks like other, three, another.
    TRUE
  } else if (kinds[[3]] == kinds[[4]] && kinds[[4]] == kinds[[5]] && 
             kinds[[1]] != kinds[[2]] && kinds[[2]] != kinds[[3]] && 
             kinds[[3]] != kinds[[1]]) {
    # This hand looks like other, another, three.
    TRUE
  } else
    FALSE
}

# Here are some pointed tests.
kinds <- c(2, 2, 5, 5, 8)
isTwoPairs(kinds)
isThreeOfAKind(kinds)
kinds <- c(4, 4, 4, 7, 9)
isTwoPairs(kinds)
isThreeOfAKind(kinds)
kinds <- c(10, 10, 10, 10, 11)
isTwoPairs(kinds)
isThreeOfAKind(kinds)
kinds <- c(9, 9, 9, 12, 12)
isTwoPairs(kinds)
isThreeOfAKind(kinds)

# Here is a random test. If you run this chunk of code repeatedly, then you 
# start to get an idea for how common two pairs and three of a kind are.
hand <- randomHand()
hand
kinds <- handKinds(hand)
kinds
isTwoPairs(kinds)
isThreeOfAKind(kinds)

# This chunk of code automates the random repetitions. Try increasing 
# numTrials, if you're willing to spend more time to (probably) get more 
# accurate results.
numTrials <- 1000000
numTwoPairs <- 0
numThreeOfAKind <- 0
for (k in 1:numTrials) {
  kinds <- handKinds(randomHand())
  if (isTwoPairs(kinds))
    numTwoPairs <- numTwoPairs + 1
  if (isThreeOfAKind(kinds))
    numThreeOfAKind <- numThreeOfAKind + 1
}
numTwoPairs / numTrials
numThreeOfAKind / numTrials

# How well do these simulation results match the probabilities that you compute 
# by hand?



### POWERBALL ###

# PowerBall is a lottery that runs in most parts of the USA. There are 10 
# levels of prizes, with the highest prize being a jackpot, whose value varies 
# over time. Here are the prizes in dollars, with a jackpot that you can set 
# yourself.
jackpot <- 20000000
payouts <- c(0, 4, 4, 7, 7, 100, 100, 50000, 1000000, jackpot)
payouts

# Here are the probabilities of winning each of those payouts. (In reality, 
# when multiple players win the jackpot, they split it. For simplicity, we 
# pretend that each jackpot winner wins the entire jackpot.)
probs <- 1 / c(1.041901, 38.32, 91.98, 701.33, 579.76, 14494.11, 36525.17, 
               913129.18, 1168805352, 292201338)
probs

# How much does PowerBall pay, on average? One way to estimate the answer is to 
# simulate a bunch of plays and average the results. Again, try increasing 
# numPlays if you want more accurate results.
numPlays <- 100000000
mean(sample(payouts, numPlays, replace=TRUE, prob=probs))

# Soon in the course we will learn a way to calculate the average payout 
# exactly and quickly.
sum(payouts * probs)

# By the way, a PowerBall ticket costs $2.