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`