Answer to Question #224448 in Statistics and Probability for Qasim

Question

Consider the following two random experiment scenarios and answer the questions that follow:

Scenario 1

A basket contains 4 red and 5 green balls. A random experiment is performed such that a ball is randomly drawn from the basket, replaced back in the basket and then again a ball is randomly drawn from the basket. Let X is a random variable containing the count of red colored balls at the end of two trials.

Scenario 2

A basket contains 6 red and 2 green balls. A random experiment is performed such that a ball is randomly drawn from the basket, not replaced back in the basket and then again a ball is randomly drawn from the basket. Let X is a random variable containing the count of green colored balls at the end of two trials.

Now answer the following questions.

.

Answer 1

Consider the following two random experiment scenarios and answer the questions that follow:

Scenario 1

A basket contains 4 red and 5 green balls. A random experiment is performed such that a ball is randomly drawn from the basket, replaced back in the basket and then again a ball is randomly drawn from the basket. Let X is a random variable containing the count of red colored balls at the end of two trials.

Scenario 2

A basket contains 6 red and 2 green balls. A random experiment is performed such that a ball is randomly drawn from the basket, not replaced back in the basket and then again a ball is randomly drawn from the basket. Let X is a random variable containing the count of green colored balls at the end of two trials.

Now answer the following questions.

a) Compare the two scenarios by showing probability tree diagrams of the two.

b) Examine and comment on the shape of the distribution for the two scenarios.

Scenario 1:

A basket contains 3 red and 4 green balls. A random experiment is performed such that a ball is randomly drawn from the basket, replaced back in the basket and then again a ball is randomly drawn from the basket.

So, the probability tree diagram for this:

Scenario 2:

A basket contains 3 red and 4 green balls. A random experiment is performed such that a ball is randomly drawn from the basket, not replaced back in the basket and then again a ball is randomly drawn from the basket.

So, the probability tree diagram for this:

a) Comparison of the two scenarios by using probability tree diagrams:

As it is given above and we can see the probability diagram of both the scenarios. So, in first scenario balls are replaced back in the basket so that in second draw the no. of balls are same as in first draw, but in second scenario there is no replacement so in second draw there is one ball less than the first draw.

b)Distribution in both scenarios is the binomial distribution with parameters n and p.

The shape of the distribution depends upon the following criteria:

- When the probability of success is less than 0.5, p<0.5, and the number of trials is small, that is, n<30, then the distribution will be positively skewed.

- When the probability of success is greater than 0.5, p>0.5, and the number of trials is small, n<30, then the distribution will be negatively skewed.

- When the probability of success is equal to 0.5, p=0.5, and the number of trials is either large or small, then the distribution will be symmetrical.

Considering the 1st scenario, where n is 2 (small) and p is 0.1837 (less than 0.5). Therefore, the shape of the distribution in this scenario is positively skewed. So, the peak of the distribution is on the left side.

Considering the 2nd scenario, where n is 2 (small) and p is 0.2857 (less than 0.5). Therefore, the shape of the distribution in this scenario is positively skewed. So, the peak of the distribution is on the left side.