According to a survey, 40% of adults watch a particular television show. Six adults are selected at random. The binomial distribution below shows the probability that k out of 6 adults watch the television show.

P (k = 0) = 0.0467

P (k = 1) = 0.1866

P (k = 2) = 0.3110

P (k = 3) = 0.2765

P (k = 4) = 0.1382

P (k = 5) = 0.0369

The distribution is

a) symmetric.

b) skewed.

c) both symmetric and skewed.

The sum of the probabilities is:

The distribution is Skewed.

The Sum of the Probabilities in the distribution is 1

The distribution is

b) skewed.

The sum of the probabilities is:

1

Step-by-step explanation:

In a binomial distribution, p represents the probability of success. Success in the sense that the event of interest happens. In the model presented, the probability of success p is 0.4 since we are informed that 40% of adults watch a particular television show.

The next quantity of significance in a binomial model is the number of independent trials, n. In our case there are 6 independent trials since we are told that 6 adults were selected at random. If we let the random variable K denote the number of adults out of the 6 who watch the television show, then K is a binomial random variable with parameters;

n = 6 and p = 0.4

A binomial distribution is only symmetric when either p is 0.5 or n is large. In the presented scenario none of this conditions is met since p is 0.4 while n is just 6 which is relatively small. Thus we conclude that the distribution is not symmetric but rather skewed.

The sum of the probabilities is any discrete probability distribution such as the bernoulli, binomial, negative binomial, poisson, or the geometric distribution is always equal to 1. That's a rule of thumb.