Discrete Probability & Expected Values

A discrete probability distribution is a statistical distribution that describes the likelihood of outcomes of a discrete random variable, where the variable can take on a countable number of values.

The expected value is the long-term average or mean of a random variable, calculated as the sum of all possible values, each multiplied by its probability of occurrence.

To calculate the expected value, multiply each possible value of the random variable by its probability and sum all these products.

P(X >= k) = 1 - P(X < k) = 1 - Σ[P(X = i)] for i = 0 to k-1.

A probability distribution must satisfy two conditions: all probabilities must be between 0 and 1, and the sum of all probabilities must equal 1.

The mean of a probability distribution is the expected value, representing the average outcome of a random variable.

Multiply each number of children by its probability: (1*0.30) + (2*0.43) + (3*0.27) = 1.97.

Use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k). For k=4, n=5, p=0.5, it equals 0.1875.

A binomial experiment is a statistical experiment that has the following properties: a fixed number of trials, two possible outcomes (success or failure), constant probability of success, and independent trials.

A discrete random variable can take on a countable number of values, while a continuous random variable can take on an infinite number of values within a given range.