Question

the table below gives the values of \\(p(x)\\) for the binomial distribution when \\(n = 6\\) and \\(p = 0.25\\).

\\(\

$$\begin{array}{|c|c|c|c|c|c|c|c|}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\\\hline p(x) & 0.178 & 0.356 & 0.297 & 0.132 & 0.033 & 0.004 & 0.000 \\\\\\hline\\end{array}$$

\\)

(a) construct the probability histogram for a binomial random variable \\(x\\) with \\(n = 6\\) and \\(p = 0.75\\). use the results from above; do not recalculate all the probabilities.

Explanation:

Analyze the symmetry of binomial distributions

Using the Binomial Probability Distribution knowledge point
Let \(X\) be a binomial random variable with parameters \(n = 6\) and \(p = 0.25\). Let \(Y\) be a binomial random variable with parameters \(n = 6\) and \(q = 1 - p = 0.75\).
The probability mass function of a binomial distribution is given by:

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

For \(Y\) with success probability \(0.75\):

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

By substituting \(j = 6 - k\), we find:

$$ P(Y = k) = \binom{6}{6-j} (0.75)^{6-j} (0.25)^j = \binom{6}{j} (0.25)^j (0.75)^{6-j} = P(X = 6-k) $$

Thus, the probability distribution for \(p = 0.75\) is the horizontal reflection of the distribution for \(p = 0.25\).

Determine the new probability values

Using the Binomial Probability Distribution knowledge point
We map the given values \(P(X = k)\) for \(p = 0.25\) to \(P(Y = 6-k)\) for \(p = 0.75\):

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Construct the probability table

Using the Binomial Probability Distribution knowledge point
We summarize the probability distribution for \(n = 6\) and \(p = 0.75\) in a table:

\(y\)	0	1	2	3	4	5	6

Answer:

To construct the probability histogram for a binomial random variable with \(n = 6\) and \(p = 0.75\), we use the binomial symmetry property:

$$ P(x \text{ successes with } p = 0.75) = P(6 - x \text{ successes with } p = 0.25) $$

The resulting probability distribution table is:

\(x\)	0	1	2	3	4	5	6