A random variable is a variable that can take on different values with certain probabilities, while a probability distribution describes the likelihood of each possible value of the random variable occurring.
Examples of random variables:
1. The outcome of rolling a six-sided die (can be 1, 2, 3, 4, 5, or 6).
2. The number of heads obtained when flipping a fair coin multiple times.

Examples of probability distributions:
1. The uniform distribution, where each outcome of rolling a fair six-sided die has an equal probability of 1/6.
2. The binomial distribution, which describes the probability of obtaining a certain number of successes (e.g., heads) in a fixed number of independent Bernoulli trials (e.g., coin flips).

A discrete variable can only take on specific, distinct values, while a continuous variable can take on any value within a certain range.

a. The length of time to get a haircut:
This is a continuous variable because it can take on any value within a range (e.g., 30 minutes, 45 minutes, 1 hour), and it's possible to have fractional values (e.g., 37.5 minutes).

b. The number of cars a jogger passes each morning while running:
This is a discrete variable because the number of cars passed can only take on whole, distinct values (e.g., 0 cars, 1 car, 2 cars), and it's not possible to pass a fractional number of cars.