# Design a binomial experiment. You will need three things: 1. The trial: it should only have two outcomes, such as “heads or tails,” “would I make it to work on time or not,” or “students who pass or fail Math 2,” etc. You also need to decide how many trials you would do. 2. The theoretical percentage of success for each trial: you may not know this, so give me a good, educated guess. 3. What’s the probability of “x times” happening? Now calculate the mean, the standard deviation, and the probability for one example of “x.” My example is as follows: 1. The trial: Finding a parking spot in Ontario Mills (a mall) right in front of the theater on a Sunday. I’m going there five times. ("Success" means you found a spot right in front of the theater.) 2. The theoretical percentage of success each trial: 20%. (That's my guess of finding a parking spot in front of the theater on any given Sunday.) What’s the percentage of "Success" all 5 times? From Table A-1, n=5 (total trials), p=0.2 (theoretical probability), x=5 (how many times I want to see it happen), the probability is 0.00032—which means that my chance of finding a spot to park in front of the theater on a Sunday all five times is 3/10000. You are required to design a binomial experiment and use your design to calculate the following:

1) Mean

2) Standard deviation

3) Probability for an example of X.

Using your example, we can define the following items.

1) Trial: Finding a parking spot in Ontario Mills right in front of...

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You are required to design a binomial experiment and use your design to calculate the following:

1) Mean

2) Standard deviation

3) Probability for an example of X.

Using your example, we can define the following items.

1) Trial: Finding a parking spot in Ontario Mills right in front of the theater on a Sunday. (Success means that you do find this parking spot.)

2) Number of trials: 5

3) Probability of success: 0.2

While you have used the Binomial Table to answer the third part of your question, the same can be achieved using the theoretical probability distribution function (pdf) of the Binomial distribution.

Given that a random variable X is binomially distributed, its pdf is given by

f(x) = nC_{x}p^{x}(1-p)^{n-x} for x = 0, 1, 2, \ldots, n where n is the number of trials and p is the probability of success.

Using this pdf, the mean E(X) of the binomial distribution is np and its variance is np(1-p) (refer to the following link for information on how to derive these values <mean, variance>).

Therefore,

1) The mean is

E(X) = np = 5 \cdot 0.2 = 1

2) The standard deviation (sd) is

sd = \sqrt{\text{Variance}} = \sqrt{np(1-p)} = \sqrt{1(1-02)} = \sqrt{0.8} = 0.8944 (to 4 decimal places).

3) P(X = 5) can be calculated from the pdf as follows:

f(x) = nC_{x} p^{x} (1-p)^{n-x} = 5C_{5} 0.2^{5} 0.8^{0} = 0.00032

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