Probability, Random Variables, and Probability Distributions

Requires a smaller overall sample size while maintaining high precision levels.

Provides results that can easily be generalized to other populations without adjustment.

Eliminates bias by including every member from each stratum in the survey.

Ensures proportional representation from each age group for more accurate subgroup analysis.

$\frac{j}{j-r}$

$\frac{j}{r-j}$

$\frac{j}{j+r}$

$\frac{g}{g+y} \times \frac{g-1}{g+y-1}$

H;I;HT;TH;TT

I;J;Z;36

H;T

I;Z

$\frac{x+x}{x+y}$

$\frac{x}{x+y}$

$\frac{x}{y} \times \frac{x-1}{y-1}$

$\frac{x}{x+y} \times \frac{x-1}{x+y-1}$

$P(A \text{ and } B) = P(A) \times P(B)$

$P(A \text{ or } B) = P(A) + P(B)$

$P(A \mid B) = P(B \mid A)$

$P(A) + P(B) = 1$

$(\frac{4}{15}) \times (\frac{4}{15})$

$(\frac{6}{15}) \times (\frac{5}{14})$

$(\frac{4}{24}) \times (\frac{4}{24})$

$(\frac{4}{15}) + (\frac{4}{15})$

Outliers can significantly increase the calculated standard deviation leading to wider confidence intervals.

Outliers are often excluded, which may result in underestimated standard deviation and misleadingly narrow confidence intervals.

Outliers don't affect the standard deviation since it's a measure of central tendency, not variability.

Outliers typically decrease the calculated standard deviations resulting in narrower confidence intervals.

$\frac{2}{3}$

\frac{12}

$\frac{1}{3}$

\frac{21}

The total number of outcomes in the sample space minus the number of outcomes in the event.

The number of outcomes in the event multiplied by the total number of outcomes in the sample space.

The number of outcomes in the event divided by the total number of outcomes in the sample space.

The number of outcomes in the event plus the total number of outcomes in the sample space.

0

1

10

2