Welcome back to Professor Baker's Math Class! In this session, we tackled Sections 6.3 and 6.4, moving deeper into the fascinating world of probability and introducing the powerful counting techniques that help us calculate the odds of winning the lottery or guessing a passcode. Here is a breakdown of the key concepts and examples we covered.

Section 6.3: Multiplication of Probability

When we look at multiple events happening in sequence (like flipping a coin AND rolling a die), we are dealing with the Multiplication Rule. The most critical step is determining if the events are independent or dependent.

Independent Events

Two events are independent if the outcome of the first does not affect the outcome of the second. For these, we simply multiply their individual probabilities:

$$P(A \cap B) = P(A) \cdot P(B)$$

Class Example: We looked at the probability of flipping a tail on a coin, rolling a 5 on a die, and drawing a Jack of Clubs from a deck. Since none of these influence each other, the math is straightforward:

$$(\frac{1}{2}) \cdot (\frac{1}{6}) \cdot (\frac{1}{52}) = \frac{1}{624} \approx 0.0016$$

Dependent Events & Conditional Probability

When the first event changes the available options for the second event (like drawing cards without replacement), the events are dependent. We must use Conditional Probability, denoted as $P(B|A)$ (read as "the probability of B given A").

$$P(A \cap B) = P(A) \cdot P(B|A)$$

Class Example: Drawing a King and then a Queen from a deck without replacement. Since we kept the King, there are only 51 cards left for the second draw:

$$(\frac{4}{52}) \cdot (\frac{4}{51}) = \frac{1}{663}$$

Section 6.4: Counting Principles, Permutations, and Combinations

Sometimes, counting the total number of outcomes is too difficult to do manually. We use three specific tools to help us count efficiently.

1. The Fundamental Counting Principle

If you have several tasks to perform in a row, you simply multiply the number of ways to do each task. We saw this with the License Plate example in Connecticut (2 letters followed by 5 numbers):

$$26 \cdot 26 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 67,600,000 \text{ combinations}$$

2. Permutations vs. Combinations

The most important question to ask yourself in Section 6.4 is: Does order matter?

  • Permutations ($nPr$): Order MATTERS. A code of 1-2-3 is different from 3-2-1.
    • Example: A lock code or determining Gold, Silver, and Bronze winners in a race.
    • Formula: $_nP_r = \frac{n!}{(n-k)!}$
  • Combinations ($nCr$): Order DOES NOT MATTER. A pizza with pepperoni and mushroom is the same as a pizza with mushroom and pepperoni.
    • Example: Lottery numbers. In the Palmetto Cash 5, you just need to match the numbers; the order doesn't matter.
    • Formula: $_nC_r = \frac{n!}{(n-k)!k!}$

We wrapped up by calculating the odds of winning the Powerball. By calculating the combinations of white balls ($69 C 5$) multiplied by the red ball options ($26$), we found the total sample space is over 292 million possibilities! This is why the probability of winning is $\frac{1}{292,201,338}$.

Keep practicing those calculator functions for $nPr$ and $nCr$, and remember to always ask yourself if the order of items changes the outcome!