University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

95

Pages:

4
Author:

Niklas

Random Variable (X): A variable that takes on different values based on the outcomes of an experiment. Sample Space (S): The set of all possible outcomes of an experiment. Probability Model: A mathematical model that assigns probabilities to each outcome in the sample space. Discrete Random Variable: A random variable with a finite or countable number of possible values. Continuous Random Variable: A random variable with an infinite number of possible values. Example: Number of Heads on Two Flips of a Coin X = number of tails in flipping a coin two times X can take on values 0, 1, or 2. The probability distribution of X is given in a table, showing the possible outcomes and their corresponding probabilities. Example: Tails on n Flips of a Coin X = number of tails in flipping a coin n times X can take on values 0, 1, 2, ..., n. There are 2^n possible outcomes. Given a fair coin, each outcome has a probability of (1/2)^n. Binomial Distribution A random variable X follows the binomial distribution with parameters n and p if: X ∈ {0, 1, ..., n} P(X = x) = C(n, x) * p^x * (1-p)^(n-x) X~Binom(n, p) is the number of successes in n independent trials with success probability p. The number of observations/trials n is fixed. The observations are independent. Each observation can be a success or a failure. The probability of success is the same for all observations. Example: Blood Types n = 5 p = 0.25 (25%) q = 1 - p = 0.75 (75%) Probability Calculations: P(X = 2) = C(5, 2) * (0.25)^2 * (0.75)^3 ≈ 0.26 P(X > 4) = P(X = 5) = C(5, 5) * (0.25)^5 * (0.75)^0 ≈ 0.001 Problem: A small airline accepts reservations for a flight with 20 seats. 90% of people who book a trip show up. The company accepts more than 20 reservations (overbooking) with the hope that no more than 20 people show up. Parts: a) If there are 20 reservations, what is the probability that the flight will be full? b) If 22 bookings have been accepted, what is the probability that some passengers who have booked will stay ashore? c) How many bookings can the airline accept if they want the probability of a passenger being left ashore to be less than 15%? Solution: a) X ~ Binom(20, 0.9) P(X = 20) = C(20, 20) * 0.9^20 * (1-0.9)^0 = 0.12 b) P(X > 20) = P(X = 21) + P(X = 22) P(X = 21) = C(22, 21) * 0.9^21 * (1-0.9)^1 ≈ 0.109 P(X = 22) = C(22, 22) * 0.9^22 * (1-0.9)^0 ≈ 0.098 P(X > 20) ≈ 0.109 + 0.098 ≈ 0.207 c) We need to find the smallest n such that P(X > n) < 0.15 By trial and error, we find that n = 21 satisfies this condition.