5.3 Counting Principles

Counting principles and permutations are essential tools in probability. They help us calculate the number of possible outcomes in various scenarios, from simple everyday choices to complex statistical problems.

Combinations and binomial probability build on these concepts. They allow us to determine the likelihood of specific events occurring, such as getting a certain number of successes in a series of trials. These skills are crucial for analyzing real-world data and making informed decisions.

Fundamental Counting Principle and Permutations

Fundamental Counting Principle application

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• States that if there are $n_1$ ways to do something and $n_2$ ways to do another thing, then there are $n_1 \times n_2$ ways to do both things
  • Useful for determining the total number of possible outcomes in a given situation
  • Helps to break down complex counting problems into simpler, more manageable parts
• Example: If a restaurant offers 5 appetizers, 8 main courses, and 3 desserts, there are $5 \times 8 \times 3 = 120$ possible three-course meals (assuming one item from each category)

Permutations and combinations in probability

• Permutation: An arrangement of objects in a specific order
  • Number of permutations of $n$ objects taken $r$ at a time denoted as $[P(n,r)](https://www.fiveableKeyTerm:p(n,r))$ or $_{n}P_{r}$
  • Formula: $P(n,r) = \frac{[n!](https://www.fiveableKeyTerm:n!)}{(n-r)!}$
  • Used when the order of selection matters (arranging books on a shelf)
• Permutations with repetition: If an object can be chosen more than once, the number of permutations is $n^r$
  • Example: A 4-digit PIN code using digits 0-9 (repetition allowed) has $10^4 = 10,000$ possible permutations
• Combination: A selection of objects without regard to the order
  • Number of combinations of $n$ objects taken $r$ at a time denoted as $[C(n,r)](https://www.fiveableKeyTerm:c(n,r))$, $_{n}C_{r}$, or $\binom{n}{r}$
  • Formula: $C(n,r) = \frac{n!}{r!(n-r)!}$
  • Used when the order of selection does not matter (selecting a committee from a group of people)

Permutations vs combinations

• Permutations: Order matters
  1. Arrange objects in a specific order
  2. Use the permutation formula: $P(n,r) = \frac{n!}{(n-r)!}$
  3. Example: Arranging the letters A, B, and C in different orders (ABC, ACB, BAC, BCA, CAB, CBA)
• Combinations: Order does not matter
  1. Select objects without regard to the order
  2. Use the combination formula: $C(n,r) = \frac{n!}{r!(n-r)!}$
  3. Example: Choosing 2 toppings from a set of 5 toppings for a pizza (choosing pepperoni and mushroom is the same as choosing mushroom and pepperoni)

Combinations and Binomial Probability

Binomial Probability Formula calculations

• Calculates the probability of exactly $x$ successes in $n$ independent trials, each with success probability $p$
  • Formula: $P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$
  • $X$: Random variable representing the number of successes
  • $n$: Number of trials
  • $p$: Probability of success on each trial
  • $x$: Number of successes
• Assumptions for the binomial probability formula:
  1. Fixed number of trials $n$
  2. Independent trials
  3. Two possible outcomes per trial (success or failure)
  4. Constant probability of success $p$ for each trial
• Example: Probability of getting exactly 3 heads in 5 coin tosses (fair coin, $p=0.5$)
  • $P(X=3) = \binom{5}{3} (0.5)^3 (1-0.5)^{5-3} = 0.3125$