Question

(b) do you see a relationship between the binomial distributions when \\(n = 6\\) for \\(p = 0.25\\) and \\(p = 0.75\\)? what is it?

• the probabilities and the probability histogram for the binomial distribution when \\(n = 6\\) and \\(p = 0.75\\) are identical to the probabilities and probability histogram when \\(n = 6\\) and \\(p = 0.25\\)
• the probabilities and the probability histogram for the binomial distribution when \\(n = 6\\) and \\(p = 0.75\\) are skewed right while the probabilities and probability histogram when \\(n = 6\\) and \\(p = 0.25\\) are symmetric.
• the probabilities and the probability histogram for the binomial distribution when \\(n = 6\\) and \\(p = 0.75\\) are a mirror image of the probabilities and probability histogram when \\(n = 6\\) and \\(p = 0.25\\).
• the probabilities and the probability histogram for the binomial distribution when \\(n = 6\\) and \\(p = 0.75\\) are symmetric while the probabilities and probability histogram when \\(n = 6\\) and \\(p = 0.25\\) are skewed left.
• the probabilities and the probability histogram for the binomial distribution when \\(n = 6\\) and \\(p = 0.75\\) and the probabilities and probability histogram when \\(n = 6\\) and \\(p = 0.25\\) show no noticeable relationship.

you may need to use the appropriate appendix table or technology to answer this question.

Explanation:

Analyze the binomial parameters

Using the Binomial Probability Distribution knowledge point
We are comparing two binomial distributions with the same number of trials \(n = 6\):

• Distribution 1: \(p_1 = 0.25\) (probability of success) and \(q_1 = 1 - p_1 = 0.75\) (probability of failure)
• Distribution 2: \(p_2 = 0.75\) (probability of success) and \(q_2 = 1 - p_2 = 0.25\) (probability of failure)

Compare probability formulas

Using the Binomial Probability Distribution knowledge point
The probability of getting \(k\) successes in \(n\) trials is given by:

$$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$

For the first distribution with \(p = 0.25\):

$$ P(X_1 = k) = \binom{6}{k} (0.25)^k (0.75)^{6-k} $$

For the second distribution with \(p = 0.75\):

$$ P(X_2 = 6-k) = \binom{6}{6-k} (0.75)^{6-k} (0.25)^k $$

Since \(\binom{6}{k} = \binom{6}{6-k}\), we have:

$$ P(X_1 = k) = P(X_2 = 6-k) $$

Determine the geometric relationship

Using the Binomial Probability Distribution knowledge point
The relation \(P(X_1 = k) = P(X_2 = 6-k)\) means that the probability of \(k\) successes in the first distribution is exactly equal to the probability of \(6-k\) successes in the second distribution.
Geometrically, this reverses the order of the probabilities along the horizontal axis.
Therefore, the probability histogram for \(p = 0.75\) is a perfect mirror image of the histogram for \(p = 0.25\).

Answer:

• (A) The probabilities and the probability histogram for the binomial distribution when \(n = 6\) and \(p = 0.75\) are identical to the probabilities and probability histogram when \(n = 6\) and \(p = 0.25\).
• (B) The probabilities and the probability histogram for the binomial distribution when \(n = 6\) and \(p = 0.75\) are skewed right while the probabilities and probability histogram when \(n = 6\) and \(p = 0.25\) are symmetric.
• (C) The probabilities and the probability histogram for the binomial distribution when \(n = 6\) and \(p = 0.75\) are a mirror image of the probabilities and probability histogram when \(n = 6\) and \(p = 0.25\). (Correct answer)
• (D) The probabilities and the probability histogram for the binomial distribution when \(n = 6\) and \(p = 0.75\) are symmetric while the probabilities and probability histogram when \(n = 6\) and \(p = 0.25\) are skewed left.
• (E) The probabilities and the probability histogram for the binomial distribution when \(n = 6\) and \(p = 0.75\) and the probabilities and probability histogram when \(n = 6\) and \(p = 0.25\) show no noticeable relationship.