STA112 – PROF. A. O. ADEJUMO

STA112: Probability 1

Course Content

  • Topics Covered:

    • Permutation and combination

    • Concepts and principles of probability

    • Random variables

    • Probability and distribution functions

    • Basic distributions: Binomial, geometric, Poisson, normal and sampling distributions

    • Exploratory data analysis

Introduction to Probability

  • Real World Applications:

    • Probability is utilized in various fields like agriculture (chance of seed germinating), road safety (accident probabilities), and politics (election outcomes).

    • Understanding probability helps assess the degree of uncertainty mathematically.

Definitions of Key Terms

  • Experiment: A process that yields a result or observation.

  • Outcome: A specific result of an experiment.

  • Sample Space (S): The set of all possible outcomes of an experiment.

  • Sample Point: Each individual outcome within the sample space.

  • Event: A subset of the sample space. If A is an event, then n(A) represents the count of sample points in A.

Probability

  • Definition: The probability of an event quantifies the likelihood of that event occurring.

  • Probability Formula: For an experiment with a finite number of equally likely outcomes,

    • P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}

    • Where:

    • n(A)n(A) = number of favorable outcomes for event A

    • n(S)n(S) = total number of outcomes in the sample space S

Examples of Probability

Example 1: Rolling a Die

  • Sample Space (S): S=1,2,3,4,5,6S = {1, 2, 3, 4, 5, 6}

  • Number of Elements: n(S)=6n(S) = 6

Example 2: Tossing a Coin Twice

  • Sample Space (S): S=HH,HT,TH,TTS = {HH, HT, TH, TT}

  • Number of Elements: n(S)=4n(S) = 4

Example 3: Tossing a Coin Thrice

  • Sample Space (S): S=HHH,HHT,HTH,THH,HTT,THT,TTH,TTTS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

  • Number of Elements: n(S)=8n(S) = 8

Example 4: Rolling Two Dice

  • Total Outcomes (n(S)): 36

  • Sum Ranges:

    • Minimum (1,1): 2

    • Maximum (6,6): 12

  • Summation Table:

    • xx: 2 3 4 5 6 7 8 9 10 11 12

    • n(x)n(x): 1 2 3 4 5 6 5 4 3 2 1

    • Total: 36

Permutation

  • Definition: Arrangement of a group of objects in order, where different arrangements of the same objects are counted separately.

  • Key Concepts: Arrangements depend on order.

  • Formula: For arranging n objects, the number of permutations is given by:

    • n!=n×(n−1)×(n−2)×⋯×1n! = n \times (n-1) \times (n-2) \times \cdots \times 1

Factorial Examples

  • 0!=10! = 1

  • 1!=11! = 1

  • 2!=22! = 2

  • 3!=63! = 6

  • 4!=244! = 24

  • 5!=1205! = 120

  • 6!=7206! = 720

Example: Seating Arrangement

  • Problem: In how many ways can three people be seated on 6 seats?

  • Solution: 6P3=6!(6−3)!=6×5×4=1206P3 = \frac{6!}{(6-3)!} = 6 \times 5 \times 4 = 120

Example: Arranging Letters in 'Economics'

  • Distinct Arrangements: Total letters = 9

    • o = 2

    • c = 2

  • Arrangements: 9!2!×2!=90720\frac{9!}{2!\times2!} = 90720

Combination

  • Definition: The selection of objects where the order does not matter.

  • Combination Formula: The number of combinations of n objects taken r at a time is denoted by (nr)\binom{n}{r}.

Example: Selecting Students

  • Problem: Select 3 students from 5.

  • Solution: (53)=10\binom{5}{3} = 10

Example: Examination Questions

  • Selection of Questions:
    -

Course Content

  • Topics Covered:

    • Permutation and combination

    • Concepts and principles of probability

    • Random variables

    • Probability and distribution functions

    • Basic distributions: Binomial, geometric, Poisson, normal and sampling distributions

    • Exploratory data analysis

Introduction to Probability

  • Real World Applications:

    • Probability is utilized in various fields like agriculture (e.g., predicting the chance of seed germinating under different conditions), road safety (assessing accident probabilities based on traffic patterns), and politics (forecasting election outcomes based on polling data).

    • Understanding probability provides tools for assessing uncertainty, enabling individuals and organizations to make informed decisions based on statistical data and risk assessments.

Definitions of Key Terms

  • Experiment: A process that yields a result or observation. This could be a single event or a repeated trial (e.g., rolling a die multiple times).

  • Outcome: A specific result of an experiment (e.g., rolling a 4 on a die). Each outcome is a distinct instance of an event happening.

  • Sample Space (S): The set of all possible outcomes of an experiment (e.g., for a single roll of a die, the sample space would be S = {1, 2, 3, 4, 5, 6}).

  • Sample Point: Each individual outcome within the sample space (e.g., the number 3 is a sample point in the experiment of rolling a die).

  • Event: A subset of the sample space. If A is an event, n(A) represents the count of sample points in A. For instance, if A is the event of rolling an even number, then A = {2, 4, 6} and n(A) = 3.

Probability

  • Definition: The probability of an event quantifies the likelihood of that event occurring, ranging from 0 (impossible event) to 1 (certain event).

  • Probability Formula: For an experiment with a finite number of equally likely outcomes:
    P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}

    • Where:

    • n(A)n(A) = number of favorable outcomes for event A

    • n(S)n(S) = total number of outcomes in the sample space S

Examples of Probability
Example 1: Rolling a Die

  • Sample Space (S): S=1,2,3,4,5,6S = {1, 2, 3, 4, 5, 6}

  • Number of Elements: n(S)=6n(S) = 6

  • For event A (rolling an even number), n(A) = 3 (i.e., {2, 4, 6}), thus P(A)=36=12P(A) = \frac{3}{6} = \frac{1}{2}.

Example 2: Tossing a Coin Twice

  • Sample Space (S): S=HH,HT,TH,TTS = {HH, HT, TH, TT}

  • Number of Elements: n(S)=4n(S) = 4

  • Event B (at least one head) has n(B) = 3, leading to P(B)=34P(B) = \frac{3}{4}.

Example 3: Tossing a Coin Thrice

  • Sample Space (S): S=HHH,HHT,HTH,THH,HTT,THT,TTH,TTTS = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

  • Number of Elements: n(S)=8n(S) = 8

  • Event C (exactly two heads) has n(C) = 3, leading to P(C)=38P(C) = \frac{3}{8}.

Example 4: Rolling Two Dice

  • Total Outcomes (n(S)): 36

  • Sum Ranges:

    • Minimum (1,1): 2

    • Maximum (6,6): 12

  • Summation Table:

    • xx: 2 3 4 5 6 7 8 9 10 11 12

    • n(x)n(x): 1 2 3 4 5 6 5 4 3 2 1

  • Total: 36

  • For event D (sum of 7), n(D) = 6, thus P(D)=636=16P(D) = \frac{6}{36} = \frac{1}{6}.

Permutation

  • Definition: Arrangement of a group of objects in order, where different arrangements of the same objects are counted separately.

  • Key Concepts: Arrangements depend on order, which distinguishes permutations from combinations.

  • Formula: For arranging n objects, the number of permutations is given by:
    n!=n×(n−1)×(n−2)×⋯×1n! = n \times (n-1) \times (n-2) \times \cdots \times 1

  • Factorial Examples:

    • 0!=10! = 1

    • 1!=11! = 1

    • 2!=22! = 2

Example: Seating Arrangement

  • Problem: In how many ways can three people be seated on 6 seats?

  • Solution: 6P3=6!(6−3)!=6×5×4=1206P3 = \frac{6!}{(6-3)!} = 6 \times 5 \times 4 = 120

Example: Arranging Letters in 'Economics'

  • Distinct Arrangements: Total letters = 9 - o = 2 - c = 2

  • Arrangements: 9!2!×2!=90720\frac{9!}{2!\times2!} = 90720

Combination

  • Definition: The selection of objects where the order does not matter, which is critical in scenarios like team selection or lottery draws.

  • Combination Formula: The number of combinations of n objects taken r at a time is denoted by (nr)\binom{n}{r}.

Example: Selecting Students

  • Problem: Select 3 students from 5.

  • Solution: (53)=10\binom{5}{3} = 10

Example: Examination Questions

  • Selection of Questions: It is often necessary to choose a set number from a larger pool while ensuring a diverse representation from various topics.