7.1: Contextual background resulting in histogram

2. Types of probability
   1. Rudimentary version is percentage of occurrences of particular events relative to total possible events
      • Examples

        Exercise \(\PageIndex{auto}\)

        Probability of drawing 7 of spades

        Answer

        1 out of 52

        Exercise \(\PageIndex{2}\)

        Probability of drawing 7 of ANY suit

        Answer

        4 of 52; equivalently 1 of 13

        Exercise \(\PageIndex{3}\)

        Outcome for rolling of single die

        Answer

        1 of 6

        Exercise \(\PageIndex{4}\)

        Probability of rolling doubles on pair of dice

        Answer

        6 of 36 or equivalently 1 of 6

        Exercise \(\PageIndex{5}\)

        Likelihood of going to jail in game of Monopoly

        Answer

        (1 of 6)\(^3\) or 1 of 216

   2. Combination of sets of outcomes
      1. Union and intersection -- adding of new events that overlap \(\ldots\) but must eliminate repeat counts
      2. Conditional probability -- likelihood given prior event occurrence (i.e. quarantine following international travel)
         • Not applicable to independent random data
      3. Permutation - \(n\) different outcomes taken \(m\) at a time
         1. Use of factorial function ! \[ P^{n}_{m} = \frac{n!}{(n-m)!} \]
         2. Order of events matter in this scenario
      4. Combination - \(n\) different outcomes taken \(m\) independently ordered times
         1. Reduces number of possibilities \[ C^{n}_{m}= \frac{n!}{m!(n-m)!}\]

         2. Exercise \(\PageIndex{6}\)

            How many ways can one create a sandwich combination?

            • List of original options: 2 patties, 1 cheese, lettuce, tomato, special sauce, middle bread ...
            • Pairing options together: proteins, veggies, condiments, containment, other

            Answer

            Add texts here. Do not delete this text first.

3. Representing statistical information
   1. Sample versus population
      1. Population is the whole/complete collection (i.e. Census 2020)
      2. Sample is subset of the selected elements of the population
      3. Creates different variables with unique identifiers

         Definition: True versus sample

         True Value	Sample statistics (with inference of "true")
         \(x' =\) true mean of whole population	\(\bar{x} =\) calculated mean from sample dataset
         \(\sigma^{2} =\) true variance	\(S_{x}^{2} =\) sample variance

   2. Histogram representation
      1. Scatter of data counts as function of another variable
         1. Uniform bin widths standard for statistical analysis
         2. non-uniform bins show equal probability, but changing ranges
         3. number of bins are user specified for representation
      2. Uniform width histogram procedure
         1. Determine total range \(x_{range} = x_{max}-x_{min}\)
         2. Select \(k\) number of intervals
         3. Define interval width \(\Delta x = x_{range}/k\)
         4. count occurrences \(n_{j}\) of data in each \(\Delta x\)
         5. Verify sum of \(n_{j}\) equals \(N\) samples
         6. Plot \(n_{j}\) versus the midpoint of each \(\Delta x\) section
      3. Change which parts for uniform height? (leave for homework assignment)
   3. Frequency distribution
      1. Same as histogram but scale \(n_{j}/N\)
      2. Creates dimensionless term rather than integer count
      3. As \(N\rightarrow \infty\) and \(\Delta x \rightarrow 0), the discrete device changes to continuum of probability density function (pdf)
      4. Can use continuous form \(p(x)\) (probability as function of value or range) for further characterization
4. Probability Density Function
   1. Distribution of values for a random variable \(x\)
   2. Probability that a specific value \(x^{*}\) occurs is \(p(x^{*})\)
      1. Positive value \(p(x) >0\) everywhere
      2. Sum of all probability is unity \[ \int_{-\infty}^{+\infty}p(x) dx = 1 \] indicating that area under the pdf must equal 1
   3. Relates to probability distribution function by integrating to limits
      1. Summing probability over a range \[ P\left[x_{0} \leq x \leq x^{*}\right] = \int_{x_{0}}^{x^{*}} p(x) dx \] Recall that \(x_{0}\) and \(x^{*}\) are specific fixed values while \(x\) is the variable of integration
      2. From plotting, the \(p(x)\) will always be positive, less than 1, but its slope will have positive and negative features
      3. Plots of \(P\left[x_{0} \leq x \leq x^{*}\right]\) always have positive slope but bounded by 0 and 1