Welcome to Section 7.3 & 7.4!

Hello Mathletes! Today's lesson covers two important probability distributions: the Discrete Uniform Distribution and the Binomial Distribution. Let's break them down and build a solid understanding.

7.3: The Discrete Uniform Distribution

The Discrete Uniform Distribution is all about situations where every outcome is equally likely. Think of it like this:

  • Fair Die: When you roll a fair six-sided die, each number (1 through 6) has an equal chance of landing face up.

Mathematically, if there are $n$ possible outcomes, the probability of each outcome is simply $\frac{1}{n}$.

Example: Consider throwing a fair six-sided die. The probability distribution is as follows:

If $X$ is the random variable representing the outcome of the throw, then:

  • $P(X=1) = \frac{1}{6}$
  • $P(X=2) = \frac{1}{6}$
  • $P(X=3) = \frac{1}{6}$
  • $P(X=4) = \frac{1}{6}$
  • $P(X=5) = \frac{1}{6}$
  • $P(X=6) = \frac{1}{6}$

7.4: The Binomial Distribution

The Binomial Distribution comes into play when we have a fixed number of independent trials, each with only two possible outcomes: success or failure. Key things to remember:

  • Fixed Number of Trials (n): You perform the experiment a set number of times.
  • Independent Trials: The outcome of one trial doesn't affect the outcome of any other trial.
  • Two Outcomes: Each trial results in either success or failure.
  • Constant Probability of Success (p): The probability of success remains the same for each trial.

The probability mass function for the binomial distribution is given by:

$$P(X = x) = {n \choose x} * p^x * (1 - p)^{(n - x)}$$

Where:

  • $X$ is the number of successes
  • $x$ is the number of successes we are interested in
  • $n$ is the number of trials
  • $p$ is the probability of success on a single trial
  • ${n \choose x}$ is the binomial coefficient, representing the number of ways to choose $x$ successes from $n$ trials. It's calculated as: ${n \choose x} = \frac{n!}{x!(n-x)!}$

Example: Tossing a coin 4 times. Let's say we want to find the probability of getting exactly 3 heads. Here, $n = 4$, $x = 3$, and $p = 0.5$ (assuming a fair coin).

Expected Value, Variance, and Standard Deviation

For a binomial random variable:

  • Expected Value: $\mu = E(X) = np$
  • Variance: $\sigma^2 = V(X) = np(1 - p)$
  • Standard Deviation: $\sigma = \sqrt{V(X)} = \sqrt{np(1 - p)}$

Example: A poll says 40% of Americans believe in ghosts. We randomly select 20 people. Let's calculate:

  • $n = 20$
  • $p = 0.4$
  • $\mu = 20 * 0.4 = 8$ (We expect 8 out of 20 to believe in ghosts)
  • $\sigma^2 = 20 * 0.4 * 0.6 = 4.8$
  • $\sigma = \sqrt{4.8} \approx 2.19$

Keep practicing, and you'll master these distributions in no time! Good luck, and see you in the next lecture!