Problem:

If the first game is played in Boston, the second game is played in New York, and the third game (if it becomes necessary) is in Boston, then complete parts (i)-(v) below.

(i) Calculate the probability that the Red Sox will win the series. (ii) Construct a probability distribution for your net win (X) in the series. Calculate your expected net win (the mean of X) and the standard deviation of X. (iii) Use Excel or R to create 10,000 random values for X. Let these random values be denoted by Y. Use these Y values to estimate your expected net win by using a 95% confidence interval. Does this confidence interval contain E(X)? (iv) Construct a frequency distribution for Y. Next, use the Chi-squared goodness of fit test to verify how closely the distribution of Y has estimated the distribution of X. (v) Use your observations of parts (ii) and (iii) above to describe whether your betting strategy is favorable to you. Write a summary of your observations and analyses in the Word document.

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I have problem in part (iv) for Chi-squared goodness of fit test, it get me error and I don't know how to solve it in R.

here is my codes:

# Calculate the expected frequencies probabilities <- c(0.125, 0.375, 0.375, 0.125) # replace with your own probabilities expected_freq <- probabilities * 10000

# Generate 10,000 random values for Y set.seed(123) # for reproducibility Y <- sample(c(-1, 0, 1, 2), size = 10000, replace = TRUE, prob = probabilities)

# Create a frequency table for Y observed_freq <- table(Y)

# Example probabilities probs <- c(0.2, 0.3, 0.5)

# Rescale probabilities probs_rescaled <- probs/sum(probs)

# Check if they add up to 1 sum(probs_rescaled)

# Perform the Chi-squared goodness of fit test chisq.test(observed_freq, p = expected_freq)

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it get me this error: Error in chisq.test(observed_freq, p = expected_freq) : probabilities must sum to 1.

how should I recall it and added to 1? can you get me its code in R to solve this error?