The binomial probability distribution is a family of probability distributions with every single distribution depending on the values of n and p. Assume that x is a binomial random variable with n = 4.

1. Determine a value of p such that the probability distribution of x is symmetric.
2. Determine a value of p such that the probability distribution of x is skewed to the right.
3. Determine a value of p such that the probability distribution of x is skewed to the left.
4. Graph each of the binomial distributions you obtained in parts a, b, and c. Locate the mean for each distribution on its graph.\
5. In general, for what values of p will a binomial distribution be symmetric? Skewed to the right? Skewed to the left?

In a symmetric binomial probability distribution, the probability of successes will be equal to that of the failures. In the case of tossing a coin, the probabilitiesof heads and tails are equal, and this can be an example of a symmetric binomial probability distribution

From the definition, as it can be envisaged,the probability will be the same for both variables, and the calculation is shown below:

$\begin{array}{rcl}\text{Probability}& =& \frac{Total probability}{2}\\ & =& \frac{1}{2}\\ & =& 0.5\\ & & \end{array}$

From the above calculation, it can be envisaged that the probability is 0.5.

Whenever the skewness remains positive (to the right), it means that the value of p must be more than 0.5. As the maximum probability can be upto 1, the value will stay between 0.5 and 1.

d. On the horizontal axis, the probabilities are plotted, and on the horizontal axis, the values are plotted from 1 to 5.In this case, it is symmetric.So, the mean value, 3,shows the highest probability

On the horizontal axis, the probabilities are plotted, and on the horizontal axis, the values are plotted from 1 to 8. In this case, it is positively skewed.So, the mean value, 4,doesnot show the highest probability, but instead, 3 showsthe highest probability.

On the horizontal axis, the probabilities are plotted, and on the horizontal axis, the values are plotted from 1 to 8.In this case, it is negatively skewed.So, the mean value, 4,doesnot show the highest probability, but instead, 6 showsthe highest probability.

Here, the value of p remains 0.5, and it indicates that the mean value of x shows the highest probability.In this case, the left tail remains exactly equal to the right tail when plotted on the graph.

Here, the value of p remains above 0.5, and it indicates that the mean value of x does not show the highest probability.In this case, the right tail remains greater than the left tail when plotted on the graph.

Here, the value of p remains below 0.5, and it indicates that the mean value of x does not show the highest probability.In this case, the left tail remains greater than the right tail when plotted on the graph.