13.3 Theoretical and Experimental Probability & Compound Events

Theoretical and Experimental Probability & Compound Events

Quick Answer: On the WEST-B Mathematics subtest (Objective 0016), probability questions require applying exact rules of counting and chance. Core formulas include: Theoretical Probability ($P(A) = \frac{\text{favorable}}{\text{total}}$), Complement Rule ($P(A') = 1 - P(A)$), Fundamental Counting Principle ($n_1 \times n_2 \times \dots \times n_k$), Independent Multiplication ($P(A \text{ and } B) = P(A) \times P(B)$), Dependent Multiplication ($P(A \text{ and } B) = P(A) \times P(B|A)$), Overlapping Addition Rule ($P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$), and the 'At Least One' Rule ($P(\ge 1) = 1 - P(\text{none})$).

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1. Probability Foundations & Definitions

Probability is the mathematical measure of the likelihood that an event will occur.

+---------------------------------------------------------------------------------------------------+
|                                 PROBABILITY SCALE & TERMINOLOGY                                   |
|                                                                                                   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | TERM                  | DEFINITION                  | MATHEMATICAL NOTATION / EXAMPLE     |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Experiment            | A repeatable process that   | Tossing a coin, rolling a die,      |   |
|   |                       | yields an outcome           | drawing a card                      |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Sample Space (S)      | The set of ALL possible     | Rolling a 6-sided die:              |   |
|   |                       | distinct outcomes           | S = {1, 2, 3, 4, 5, 6}, |S| = 6     |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Event (E)             | Any subset of sample space S| Rolling an even number:             |   |
|   |                       |                             | E = {2, 4, 6}, |E| = 3              |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Probability Axiom     | Probability must fall       | 0 ≤ P(E) ≤ 1                        |   |
|   |                       | strictly between 0 and 1    | (or 0% to 100%)                     |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Impossible Event      | An event that cannot occur  | P(∅) = 0 (e.g., rolling a 7 on die) |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Certain Event         | An event guaranteed to occur| P(S) = 1 (e.g., rolling ≤ 6 on die) |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

                                   THE PROBABILITY SCALE

         Impossible             Unlikely             Even Chance           Likely               Certain
             |----------------------|---------------------|-------------------|--------------------|
            0.0                    0.25                  0.50                0.75                 1.0
            0%                     25%                   50%                 75%                  100%
            0                      1/4                   1/2                 3/4                  1

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2. Theoretical vs. Experimental Probability & Law of Large Numbers

+---------------------------------------------------------------------------------------------------+
|                       THEORETICAL VS. EXPERIMENTAL PROBABILITY MATRIX                             |
|                                                                                                   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | PROPERTY              | THEORETICAL PROBABILITY     | EXPERIMENTAL (EMPIRICAL) PROBABILITY|
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Calculation Basis     | Mathematical analysis of    | Actual collected data from trials/  |   |
|   |                       | equally likely outcomes     | experiments                         |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Formula               | P(A) = n(A) / n(S)          | P_exp(A) = (Observed Successes) /   |   |
|   |                       |                             | (Total Number of Trials)            |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Example: Fair Coin    | P(Heads) = 1/2 = 0.50       | 54 Heads in 100 flips = 54/100=0.54 |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Law of Large Numbers  | Fixed theoretical benchmark | As trials n → ∞, Experimental P     |   |
|   |                       | value                       | approaches Theoretical P            |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

The Complement Rule

The complement of an event $A$ (denoted $A'$, $A^c$, or $\text{not } A$) consists of all outcomes in sample space $S$ that are not in $A$:

$$P(A') = 1 - P(A) \iff P(A) + P(A') = 1$$

• The 'At Least One' Shortcut: When calculating the probability that an event occurs at least once across multiple trials: $$P(\text{at least one occurrence}) = 1 - P(\text{zero occurrences})$$

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3. Counting Principles, Permutations & Combinations

A. The Fundamental Counting Principle (Multiplication Rule)

If an experiment consists of $k$ consecutive sequential steps, where step 1 can occur in $n_1$ ways, step 2 in $n_2$ ways, ..., and step $k$ in $n_k$ ways, the total number of compound outcomes in the sample space is:

$$\text{Total Outcomes} = n_1 \times n_2 \times n_3 \times \dots \times n_k$$

Example: A school cafeteria offers a lunch combo with a choice of 3 sandwiches, 4 sides, and 2 drinks. Total distinct lunches $= 3 \times 4 \times 2 = 24$.

B. Permutations vs. Combinations

+---------------------------------------------------------------------------------------------------+
|                                PERMUTATION VS. COMBINATION MATRIX                                 |
|                                                                                                   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | FEATURE               | PERMUTATIONS P(n, r)        | COMBINATIONS C(n, r)                |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Does Order Matter?    | YES (Order / Position counts| NO (Order does not matter; group/   |
|   |                       | e.g., President, VP, Sec)   | committee selection)                |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Mathematical Formula  | P(n, r) = n! / (n - r)!     | C(n, r) = n! / [r! (n - r)!]        |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Example: Select 2     | P(5, 2) = 5 × 4 = 20        | C(5, 2) = (5 × 4) / (2 × 1) = 10    |
|   | items from 5 {A,B,...}| (AB is distinct from BA)    | ({A, B} is same group as {B, A})    |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

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4. Compound Events: Independent vs. Dependent

A compound event involves the simultaneous or sequential occurrence of two or more simple events.

+---------------------------------------------------------------------------------------------------+
|                                 COMPOUND PROBABILITY RULES                                        |
|                                                                                                   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | EVENT RELATIONSHIP    | CONDITION                   | PROBABILITY FORMULA                 |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Independent Events    | Outcome of Event A does NOT | P(A and B) = P(A) × P(B)            |
|   | (With Replacement)    | affect Event B              |                                     |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Dependent Events      | Outcome of Event A ALTERS   | P(A and B) = P(A) × P(B | A)        |
|   | (Without Replacement) | the sample space of Event B |                                     |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Mutually Exclusive    | Events CANNOT occur at the  | P(A or B) = P(A) + P(B)             |
|   | (Disjoint) Events     | same time: P(A and B) = 0   |                                     |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Overlapping Events    | Events CAN occur            | P(A or B) = P(A) + P(B) - P(A and B)|
|   | (Non-Disjoint)        | simultaneously              |                                     |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

                       Venn Diagram: Overlapping vs. Disjoint

          Overlapping Events (A ∩ B ≠ ∅)            Mutually Exclusive (Disjoint)
             +----------+   +----------+                +----------+   +----------+
            /   Only A   \ /   Only B   \              /            \ /            \
           |              X              |            |   Event A    |   Event B    |
            \    (A)     / \    (B)     /              \            / \            /
             +----------+   +----------+                +----------+   +----------+
                        Intersection
                         (A and B)

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5. Tree Diagrams & Geometric Probability

A. Tree Diagrams

A tree diagram visualizes multi-stage probabilistic experiments by displaying branches for each sequential outcome with conditional probabilities labeled on each branch.

                            Tree Diagram: Two Coin Flips

                                                 Outcome    Probability
                                                +-------+  +-----------+
                                           H -->|  H, H |   1/2 × 1/2 = 1/4
                                  1/2     /     +-------+  +-----------+
                                   +----> H
                                  /       \ 1/2 +-------+  +-----------+
                                 /         T -->|  H, T |   1/2 × 1/2 = 1/4
                              Start             +-------+  +-----------+
                                 \         1/2  +-------+  +-----------+
                                  \ 1/2    H -->|  T, H |   1/2 × 1/2 = 1/4
                                   +----> T     +-------+  +-----------+
                                          \ 1/2 +-------+  +-----------+
                                           T -->|  T, T |   1/2 × 1/2 = 1/4
                                                +-------+  +-----------+

B. Geometric (Area) Probability

In geometric probability, outcomes correspond to points in a geometric region (length, area, or volume):

$$P(\text{Landing in Target Region}) = \frac{\text{Area of Target Region}}{\text{Total Area of Sample Space Region}}$$

• Concentric Circles Example: If a bullseye has radius $r$ inside a dartboard of radius $R$: $$P(\text{Bullseye}) = \frac{\pi r^2}{\pi R^2} = \frac{r^2}{R^2}$$

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6. Step-by-Step Worked Probability Problems

Problem 1: Dependent Probability (Without Replacement)

Problem: A jar contains $6$ red marbles, $4$ blue marbles, and $2$ yellow marbles ($12$ marbles total). Two marbles are drawn at random one after another without replacement. What is the probability that both marbles drawn are red?

Step-by-Step Solution:

1. Calculate probability of drawing first red marble: $$P(R_1) = \frac{\text{Red Marbles}}{\text{Total Marbles}} = \frac{6}{12} = \frac{1}{2}$$
2. Calculate conditional probability of drawing second red marble:
   • One red marble has been removed: $5$ red marbles remaining.
   • Total marbles remaining: $12 - 1 = 11$ marbles.
   $$P(R_2 \mid R_1) = \frac{5}{11}$$
3. Multiply probabilities (Multiplication Rule for Dependent Events): $$P(R_1 \text{ and } R_2) = P(R_1) \times P(R_2 \mid R_1) = \frac{6}{12} \times \frac{5}{11} = \frac{1}{2} \times \frac{5}{11} = \frac{5}{22}$$
4. Convert to decimal/percentage: $$\frac{5}{22} \approx 0.2273 = 22.73\%$$

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Problem 2: Overlapping Addition Rule (Deck of Cards)

Problem: A single card is drawn at random from a standard, well-shuffled 52-card deck. What is the probability that the card drawn is either an Ace OR a Heart?

Step-by-Step Solution:

1. Identify individual event probabilities:
   • Event $A$ (Drawing an Ace): There are $4$ Aces in a deck $\implies P(A) = \frac{4}{52}$.
   • Event $B$ (Drawing a Heart): There are $13$ Hearts in a deck $\implies P(B) = \frac{13}{52}$.
2. Identify overlapping outcomes ($A \cap B$):
   • There is exactly $1$ card that is both an Ace and a Heart (the Ace of Hearts).
   $$P(A \text{ and } B) = \frac{1}{52}$$
3. Apply the Addition Rule for Overlapping Events: $$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$ $$P(A \text{ or } B) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{4 + 13 - 1}{52} = \frac{16}{52}$$
4. Simplify the fraction: $$\frac{16}{52} = \frac{4}{13} \approx 30.77\%$$

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Problem 3: The 'At Least One' Complement Problem

Problem: A couple plans to have $4$ children. Assuming each child is equally likely to be a boy or a girl ($P(\text{Boy}) = P(\text{Girl}) = 0.5$) independently, what is the probability that the couple has at least one girl?

Step-by-Step Solution:

1. Define the complementary event:
   • The complement of "at least one girl" is "zero girls" (which means all 4 children are boys).
2. Calculate probability of zero girls (all 4 boys): $$P(\text{Boy}_1 \text{ and } \text{Boy}_2 \text{ and } \text{Boy}_3 \text{ and } \text{Boy}_4) = \left(\frac{1}{2}\right)^4 = \frac{1}{16}$$
3. Apply Complement Rule: $$P(\ge 1 \text{ Girl}) = 1 - P(\text{0 Girls}) = 1 - \frac{1}{16} = \frac{16}{16} - \frac{1}{16} = \frac{15}{16}$$
4. Convert to decimal/percentage: $$\frac{15}{16} = 0.9375 = 93.75\%$$ There is a $\frac{15}{16}$ ($93.75\%$) chance of having at least one girl.