Final answer:
Geometric probability is concerned with the trials until the first success in a sequence of independent tries, with X denoting the number of trials and X following a geometric distribution G(p).
Step-by-step explanation:
Geometric probability deals with the number of trials needed for the first success in a sequence of independent Bernoulli trials (each with the same probability of success).
To answer the student's question, let the random variable X be the number of trials until the first success. The distribution of X can be expressed as X ~ G(p), where p is the probability of success on any single trial, and q (1 - p) is the probability of failure.
To find the probability of a specific outcome, such as P(x = 4), we use the formula for the geometric distribution: P(X = x) = p * (1-p)^(x-1). To find the sum of probabilities up to a certain point, such as P(x ≤ 3), we would sum the probabilities of each individual outcome: P(X = 1) + P(X = 2) + P(X = 3).
The expected value, or the average number of years a physics major might do postgraduate research, is calculated as 1/p for a geometric distribution. The expected value represents the mean of the distribution.