Class Notes: March 2, 2023 - Sections 4.8, 5.1, and 5.2

Welcome back to math class! Today's blog post summarizes the key concepts covered on March 2nd, 2023, focusing on sections 4.8, 5.1, and 5.2 from your textbook. Make sure to review these concepts thoroughly as they will be crucial for the upcoming test. Let's dive in!

Section 4.8: Factorials, Permutations, and Combinations

This section introduces fundamental counting principles. Here are the key concepts:

  • Factorial: The product of all positive integers less than or equal to a given positive integer $k$. It's denoted as $k!$. Mathematically, $k! = k \times (k-1) \times (k-2) \times ... \times 2 \times 1$. By definition, $0! = 1$. For example, $3! = 3 \times 2 \times 1 = 6$ and $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
  • Permutations: A permutation is an arrangement of objects in a specific order. The number of permutations of $r$ objects chosen from a set of $m$ objects is given by the formula: $$mP_r = \frac{m!}{(m-r)!}$$. Order matters in permutations.
  • Combinations: A combination is a selection of objects where order doesn't matter. The number of combinations of $r$ objects chosen from a set of $m$ objects is given by the formula: $$mC_r = \frac{m!}{r!(m-r)!}$$.

Example: License Plates

Consider license plates that consist of three letters followed by three digits.

  1. How many different license plates are possible?
    • There are 26 choices for each of the three letters, and 10 choices for each of the three digits. Therefore, the total number of possible license plates is $26 \times 26 \times 26 \times 10 \times 10 \times 10 = 17,576,000$.
  2. How many possibilities are there for license plates on which no letter or digit is repeated?
    • In this case, we have $26 \times 25 \times 24 \times 10 \times 9 \times 8 = 11,232,000$ possibilities.

Sections 5.1 and 5.2: Random Variables and Discrete Random Variables

These sections introduce the concept of random variables and discrete random variables, laying the groundwork for probability distributions.

  • Random Variable: A quantitative variable whose value depends on chance.
  • Discrete Random Variable: A random variable whose possible values can be listed. In particular, a random variable with only a finite number of possible values is a discrete random variable. Examples include the sum of the dice when a pair of fair dice are rolled, the number of puppies in a litter, or the return on an investment.
  • Probability Distribution: A listing of the possible values and corresponding probabilities of a discrete random variable, or a formula for the probabilities.
  • Mean of a Discrete Random Variable: The mean ($\mu$) is calculated as $\mu = \sum x_i P(x_i)$, where $x_i$ are the possible values and $P(x_i)$ are their corresponding probabilities.
  • Standard Deviation of a Discrete Random Variable: The standard deviation ($\sigma$) measures the spread of the distribution and is calculated as: $$\sigma = \sqrt{\sum (x - \mu)^2 P(X = x)}$$ or, using a computing formula, $$\sigma = \sqrt{\sum x^2 P(X = x) - \mu^2}$$.

Remember to practice these concepts with various examples to solidify your understanding. Good luck with your test preparation!