Got it, let's work through this step by step to get your simulation up and running correctly. The key pieces you're missing are defining a proper standard deck for sampling, implementing the with-replacement draw logic, and wrapping everything in a repeatable trial loop.

First, let's map a real 52-card deck to point values:

• 4 Aces (each worth 1, which we'll adjust to 11 later if needed)
• 4 cards each for 2 through 9
• 16 cards worth 10 (10, Jack, Queen, King — 4 suits each)

In R, we can define this deck with:

# Define a standard 52-card deck using point values
deck <- c(rep(1, 4), rep(2:9, each = 4), rep(10, 16))

Your handValue function has solid core logic, so we'll tweak it for readability while keeping the key checks:

handValue <- function(cards) {
  value <- sum(cards)
  
  # Adjust Ace from 1 to 11 if it doesn't make the hand bust
  if (any(cards == 1) && value <= 11) {
    value <- value + 10
  }
  
  # Return 0 for bust, 21.5 for Blackjack, else the hand value
  if (value > 21) {
    return(0)
  } else if (value == 21 && length(cards) == 2) {
    return(21.5) # Unique marker for Blackjack
  } else {
    return(value)
  }
}

The critical part here is using sample(..., replace = TRUE) to simulate drawing cards with replacement. We'll run 5000 trials with a vectorized approach for efficiency:

set.seed(5000) # Ensure reproducibility of your results
num_trials <- 5000

# Run all trials: draw 2 cards with replacement, get hand value for each
trial_results <- sapply(1:num_trials, function(x) {
  hand <- sample(deck, 2, replace = TRUE) # This is the with-replacement logic!
  handValue(hand)
})

# Count how many trials resulted in Blackjack (marked by 21.5)
blackjack_count <- sum(trial_results == 21.5)

# Calculate the relative frequency (estimated probability)
estimated_probability <- blackjack_count / num_trials

Add a simple print statement to see your estimate:

cat("Out of", num_trials, "with-replacement trials:\n")
cat("- Number of Blackjacks:", blackjack_count, "\n")
cat("- Estimated probability:", round(estimated_probability, 4), "\n")

Since Blackjack only applies to 2-card hands, you could skip the full hand value function and directly check for the valid combinations (Ace + 10-point card, in either order):

blackjack_count_simple <- sum(sapply(1:num_trials, function(x) {
  hand <- sample(deck, 2, replace = TRUE)
  (hand[1] == 1 && hand[2] == 10) || (hand[1] == 10 && hand[2] == 1)
}))

This will give you the same count, and might be easier to follow for this specific simulation.

Putting it all together:

set.seed(5000)

# Define standard deck point values
deck <- c(rep(1, 4), rep(2:9, each = 4), rep(10, 16))

# Hand value calculator
handValue <- function(cards) {
  value <- sum(cards)
  if (any(cards == 1) && value <= 11) {
    value <- value + 10
  }
  if (value > 21) {
    return(0)
  } else if (value == 21 && length(cards) == 2) {
    return(21.5)
  } else {
    return(value)
  }
}

# Run 5000 with-replacement trials
num_trials <- 5000
trial_results <- sapply(1:num_trials, function(x) {
  hand <- sample(deck, 2, replace = TRUE)
  handValue(hand)
})

# Calculate and print results
blackjack_count <- sum(trial_results == 21.5)
estimated_probability <- blackjack_count / num_trials

cat("Out of", num_trials, "with-replacement trials:\n")
cat("- Number of Blackjacks:", blackjack_count, "\n")
cat("- Estimated probability:", round(estimated_probability, 4), "\n")

内容的提问来源于stack exchange，提问作者devonwilliams064