In a binomial distribution, how does increasing the value of $n$ (the number of trials) affect the shape of the binomial probability histogram?

• A. As $n$ increases, the histogram becomes less symmetric and more irregular.
• B. As $n$ increases, the histogram becomes more skewed to the left.
• C. As $n$ increases, the histogram becomes more skewed to the right.
• D. As $n$ increases, the histogram becomes more symmetric and approaches a normal distribution.

Answer: (D)

Verified step by step guidance

1. Recall that a binomial distribution models the number of successes in \( n \) independent trials, each with probability \( p \) of success.
2. Understand that the shape of the binomial probability histogram depends on both \( n \) (number of trials) and \( p \) (probability of success).
3. As \( n \) increases, the distribution tends to become smoother and more symmetric, especially when \( p \) is not too close to 0 or 1.
4. This happens because of the Central Limit Theorem, which states that the sum (or average) of a large number of independent random variables tends to follow a normal distribution, regardless of the original distribution.
5. Therefore, increasing \( n \) makes the binomial histogram approach the bell-shaped curve of a normal distribution, becoming more symmetric and less skewed.