To construct a binomial probability distribution table with $n = 5$ (number of trials) and $p = 0.23$ (probability of success), we can calculate the probability for each number of successes $x$, where $x$ can range from 0 to 5. The formula for the binomial probability mass function is:

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$ where $\binom{n}{x}$ is the binomial coefficient, calculated as:

$\binom{n}{x} = \frac{n!}{x!(n-x)!}$

Step 1: Calculate the binomial probabilities for each $x$ from 0 to 5.

Given:

• $n = 5$
• $p = 0.23$
• $q = 1 - p = 0.77$

We calculate each probability as follows:

$P(X = x) = \binom{5}{x} (0.23)^x (0.77)^{5-x}$

$x$ (Number of Successes)	$\binom{5}{x}$	Probability $P(X = x)$
0	$\binom{5}{0} = 1$	$P(X = 0) = 1 \times (0.23)^0 \times (0.77)^5 = 0.2861$
1	$\binom{5}{1} = 5$	$P(X = 1) = 5 \times (0.23)^1 \times (0.77)^4 = 0.4268$
2	$\binom{5}{2} = 10$	$P(X = 2) = 10 \times (0.23)^2 \times (0.77)^3 = 0.2181$
3	$\binom{5}{3} = 10$	$P(X = 3) = 10 \times (0.23)^3 \times (0.77)^2 = 0.0585$
4	$\binom{5}{4} = 5$	$P(X = 4) = 5 \times (0.23)^4 \times (0.77)^1 = 0.0082$
5	$\binom{5}{5} = 1$	$P(X = 5) = 1 \times (0.23)^5 \times (0.77)^0 = 0.0005$

Step 2: Calculate the mean and standard deviation.

• Mean ($\mu$) of a binomial distribution is given by:

$\mu = n \times p = 5 \times 0.23 = 1.15$

• Standard Deviation ($\sigma$) of a binomial distribution is given by:

$\sigma = \sqrt{n \times p \times (1-p)} = \sqrt{5 \times 0.23 \times 0.77} = \sqrt{0.8855} = 0.9405$

Final Summary:

• Mean: $\mu = 1.15$
• Standard Deviation: $\sigma = 0.9405$

Would you like further clarification on these calculations or more details on any step?

Here are five related questions to deepen your understanding:

1. How would the probabilities change if $p$ were increased to 0.5?
2. What happens to the standard deviation if the number of trials, $n$, increases?
3. How would you calculate cumulative probabilities for this binomial distribution?
4. What is the probability of getting exactly 3 successes for $p = 0.4$ and $n = 7$?
5. How does the shape of a binomial distribution change as $p$ approaches 1?

Tip: For large values of $n$ and probabilities near 0.5, the binomial distribution begins to resemble the normal distribution.