Chapter 3: Basic Concepts of Probability (Week 3) - Summary Notes

Sample Spaces and Events

A random experiment is a mechanism that produces a definite outcome that cannot be predicted with certainty.
The sample space associated with a random experiment is the set of all possible outcomes.
An event is a subset of the sample space.
An event E is said to occur on a particular trial of the experiment if the outcome observed is an element of the set E.
The probability of an outcome e can be expressed as p(e). For the general formulas governing probabilities, such as the range of possible probability values and how to calculate the probability of an event from its constituent outcomes, refer to the "All Formulas and Equations" section at the end.

Examples

Sample space of flipping a coin once:
S = {H, T}.
Sample space of flipping a coin twice:
S = {HH, HT, TH, TT}.
Sample space of tossing a balanced die once:
S = {1, 2, 3, 4, 5, 6}.
Three-child family (boys = b, girls = g) sample space (each child independently either b or g):
S = {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}.
Tree example (brief): The line segments are branches; the right ending point of each branch is a node. The tree represents all possible sequences of outcomes such as after tossing a coin twice: S = {HH, HT, TH, TT}.

Complements (Sec 2.1)

The complement of an event A in a sample space S, denoted A^c, is the collection of all outcomes in S that are not in A.
It corresponds to negating the description of the event A in words.
The probability rule for complements is provided in the "All Formulas and Equations" section.

Example: Rolling a single die

Sample space: S = {1, 2, 3, 4, 5, 6}
Let E = {2, 4, 6} (even number).
Complement: E^c = {1, 3, 5}
Let T = {3, 4, 5, 6} (greater than 2).
Complement: T^c = {1, 2}

Intersections (Sec 2.2)

The intersection of events A and B, denoted A
B, is the collection of all outcomes that are elements of both A and B.
It corresponds to describing both events using the word "and".
Example description: throwing a die and getting an even number and greater than two.

Example: Rolling a die

Let E = {2, 4, 6} (even).
Let T = {3, 4, 5, 6} (greater than two).
Intersection: E
T = {4, 6}.
Probabilities: if the die is fair,P(E
T) = P(4) + P(6) = rac{1}{6} + rac{1}{6} = rac{2}{6} = rac{1}{3}.
Note: The source material shows a minor arithmetic misprint (2/6 = 2/3 is incorrect); the correct value is rac{1}{3}.

Mutually Exclusive Events (Sec 2.2)

Events A and B are mutually exclusive if they have no outcomes in common.
This means it is impossible for both A and B to occur on a single trial.
The probability rule for mutually exclusive events is provided in the "All Formulas and Equations" section.
Any event A and its complement A^c are mutually exclusive.
A and B can be mutually exclusive without being complements.

Mutually exclusive example

Let E = {2, 4, 6} (even).
Consider possible A's such as A = {3}, A = {5}, or A = {1, 3, 5}; each is mutually exclusive with E whenever they do not share outcomes.

Union of Events (Sec 2.3)

The union of events A and B, denoted A
B, is the collection of all outcomes that are in A or B or both. It corresponds to the word "or".

Example: Rolling a die

Let E = {2, 4, 6} (even).
Let T = {3, 4, 5, 6} (greater than two).
Union: E
T = {2, 3, 4, 5, 6}.
Note: 4 appears in both sets but is listed once in the union.

Additive Rule of Probability

The additive rule for any two events A and B is provided in the "All Formulas and Equations" section.

Example: Additive rule (M and E)

A tutoring service: among all students, suppose
- P(M) = 0.63 (need help in Mathematics)
- P(E) = 0.34 (need help in English)
- P(M
  E) = 0.27 (need both)
Then the probability of needing help in Mathematics or English is calculated as
P(M
E) = P(M) + P(E) - P(M
E) = 0.63 + 0.34 - 0.27 = 0.70.

Textbook Exercises (conceptual checks)

If S = {a, b, c, d} with probabilities P(a) = 0.32, P(b) = 0.17, P(c) = 0.28, P(d) = 0.23:
- Check that S is a complete sample space: P(a) + P(b) + P(c) + P(d) = 0.32 + 0.17 + 0.28 + 0.23 = 1.
- A = {b, c}
  P(A) = P(b) + P(c) = 0.17 + 0.28 = 0.45.
- B = {b, c, d}
  P(B) = P(b) + P(c) + P(d) = 0.17 + 0.28 + 0.23 = 0.68.
Complements: P(A^c) = 1 - P(A) = 1 - 0.45 = 0.55.
(Alternative: Ac = {a, d}
P(A^c) = P(a) + P(d) = 0.32 + 0.23 = 0.55.)
Intersections and unions:
- P(A
  B) = P({b, c}) = 0.45.
- P(A
  B) = P({b, c, d}) = 0.17 + 0.28 + 0.23 = 0.68.
- Alternatively, P(A
  B) = 1 - P({a}) = 1 - 0.32 = 0.68.

Conditional Probability (Sec 3.1)

The conditional probability of A given B, denoted P(A
B), is the probability that A occurs in a trial for which B is known to occur.
The definition formula is provided in the "All Formulas and Equations" section.
Note: This formula is valid whenever P(B) > 0.

Independent Events (Sec 3.2)

Events A and B are independent if their joint probability equals the product of their individual probabilities. This rule is provided in the "All Formulas and Equations" section.
If the rule holds, then A and B are independent; if not, they are not independent.

Example: Dice independence

Let A = {3} and B = {1, 3, 5} for a single fair die.
P(A) =
rac{1}{6}, P(B) =
rac{3}{6} = rac{1}{2}, P(A
B) = P(3) =
rac{1}{6}.
Product: P(A)P(B) =
rac{1}{6}
rac{1}{2} = rac{1}{12}..
Since P(A
B) = rac{1}{6}
rac{1}{12} = P(A)P(B), A and B are not independent.

Probabilities on Tree Diagrams (Sec 3.3)

Tree diagrams can simplify probability problems.
The probability of the event corresponding to any node on a tree is the product of the numbers on the unique path of branches that leads to that node from the start.
If an event corresponds to several final nodes, its probability is obtained by summing the numbers next to those nodes.

Example: Drawing two marbles without replacement

Jar contains 10 marbles: 7 black (B) and 3 white (W).
Draw two marbles without replacement.
Probabilities on the tree:
- P(B1
  B2) =
  \left(\frac{7}{10}\right) \left(\frac{6}{9}\right) = \frac{42}{90} = \frac{7}{15} \approx 0.4667 (\approx 0.47).
- P(W1
  W2) =
  \left(\frac{3}{10}\right) \left(\frac{2}{9}\right) = \frac{6}{90} = \frac{1}{15} \approx 0.0667 (\approx 0.07).
- P(exactly one black) = P(B1
  W2) + P(W1
  B2) =
  \left(\frac{7}{10}\right)\left(\frac{3}{9}\right) + \left(\frac{3}{10}\right)\left(\frac{7}{9}\right) = \frac{21}{90} + \frac{21}{90} = \frac{42}{90} = \frac{7}{15} \approx 0.4667 (\approx 0.46).
- P(at least one black) = P(B1
  B2) =
  1 - P(W1
  W2) = 1 - \frac{1}{15} = \frac{14}{15} \approx 0.9333 (\approx 0.93).

Key Terms & Takeaways

Sample space: The set of all possible outcomes of a random experiment.
Event: Any set of outcomes (a subset of the sample space).
Probability of an outcome: A number that measures the likelihood of that outcome, satisfying 0
p
1.
Probability of an event: A number that measures the likelihood of the event, obtained by summing the probabilities of its constituent outcomes.
Complement of an event: The event does not occur; its probability is related by a rule found in the "All Formulas and Equations" section.
Intersection of two events: Both events occur; symbol A
B.
Union of two events: One or the other (or both) occur; symbol A
B.
Mutually exclusive events: Events that cannot both occur on the same trial; their intersection probability is 0 (see "All Formulas and Equations").
Conditional probability: The probability of an event given that another event has occurred; definition found in "All Formulas and Equations".
Independent events: Events whose joint probability equals the product of their probabilities; rule found in "All Formulas and Equations".
Tree diagrams: Probabilities along a path multiply; final-event probabilities are sums of path-probabilities for the final outcomes.

Key Takeaways (condensed)

The sample space is the collection of all possible outcomes; an event is a subset of this space.
The probability of any outcome is between 0 and 1, and the probabilities of all outcomes sum to 1.
The probability of an event is the sum of its constituent outcome probabilities. For complementation, unions, intersections, and conditional probabilities, refer to the "All Formulas and Equations" section at the end.
Tree diagrams offer a practical method for handling sequential draws and for computing probabilities of complex events by path-products and path-sums.

All Formulas and Equations

Probability of an outcome e in the sample space S: The probability p satisfies 0 \le p \le 1.
Probability of an event A comprised of outcomes e1, e2, \ldots, e_k: P(A) = P(e1) + P(e2) + P(e3) + \cdots + P(ek).
Probability rule for complements: P(A^c) = 1 - P(A).
Probability rule for mutually exclusive events A and B: P(A
B) = 0.
Additive Rule of Probability for any two events A and B: P(A \cup B) = P(A) + P(B) - P(A
B).
Conditional probability of A given B (where P(B) > 0): P(A \mid B) = \frac{P(A \cap B)}{P(B)}.
Rule for independent events A and B: P(A \cap B) = P(A) \cdot P(B).