For each of the two situations described below, please indicate if the variable
ID: 3200185 • Letter: F
Question
For each of the two situations described below, please indicate if the variable X (as defined in each situation) can be considered a binomial random variable. If you think that X is a binomial variable, please explain how the situation specifically meets each of the three criteria, and identify the values of n and p. If you think X cannot be considered a binomial variable, please indicate which of the three criteria is/are not met (indicate all that apply), and provide a brief explanation for your choice(s).
Situation 1: A fair coin is tossed over and over again. Let X = the number of tosses until the third TAILS appears.
Situation 2: A box contains 10 marbles: 4 are red, 3 are white, and 3 are blue. A marble is randomly selected, returned to the box, then another marble is randomly selected. Let X = the number of red marbles selected in the two consecutive trials.
Explanation / Answer
Situation 1:
In this case of n repeated Bernoulli trials, we are interested the the 3rd success in the rth trial.
This means that the rth trial results in a successm and in the previous (r-1) trials, there are 2 successes.
The probability that there are 2 successes in (r-1) trials is given y:
(r-1)C(2 p2 qr-2.
The probability that rth trial is a success is p.
The probability that rth trial is 3rd success is given by:
(r-1)C2 p2 qr-2 p, where q = 1-p.
This probability distribution is, thus, not Binomial Distribution.
It is called Negative Binaminal Distribution or Pascal Distribution.
Situation 2:
This is a binomial Distribution, because the marble is returned to the box after each trial.
Let x = event of rede ball is selected.
So,
p= Probability of success = 4/10 = 0.40 and
The probibility of failure = q = 1 p = 0.60
reman the same through the experiment.
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