Find $$$P{\left(X = 5 \right)}$$$ for geometric distribution with $$$n = 5$$$ and $$$p = 0.22$$$

Calculate the various values for the geometric distribution with $$$n = 5$$$ and $$$p = 0.22 = \frac{11}{50}$$$ (include a success trial).

Mean: $$$\mu = \frac{1}{p} = \frac{1}{\frac{11}{50}} = \frac{50}{11}\approx 4.545454545454545$$$A.

Variance: $$$\sigma^{2} = \frac{1 - p}{p^{2}} = \frac{1 - \frac{11}{50}}{\left(\frac{11}{50}\right)^{2}} = \frac{1950}{121}\approx 16.115702479338843.$$$A

Standard deviation: $$$\sigma = \sqrt{\frac{1 - p}{p^{2}}} = \sqrt{\frac{1 - \frac{11}{50}}{\left(\frac{11}{50}\right)^{2}}} = \frac{5 \sqrt{78}}{11}\approx 4.014436757421749.$$$A

$$$P{\left(X = 5 \right)} = 0.0814331232$$$A

$$$P{\left(X \lt 5 \right)} = 0.62984944$$$A

$$$P{\left(X \leq 5 \right)} = 0.7112825632$$$A

$$$P{\left(X \gt 5 \right)} = 0.2887174368$$$A

$$$P{\left(X \geq 5 \right)} = 0.37015056$$$A