CBSE Class 9 Mathematics: Introduction to Probability Study Guide

Foundations and Historical Development of Probability

• Everyday language frequently employs words denoting uncertainty, including "probably," "doubt," "most likely," "chances," and "odds."
• In mathematical terms, probability is the branch of mathematics that quantitatively measures the degree of certainty or uncertainty associated with the occurrence of an event.
• While classical probability historically originated from the analysis of games of chance (such as dice and card games in seventeenth-century Europe), introductory secondary mathematics (CBSE Class 9) focuses on the empirical or experimental approach to probability.
• The experimental approach measures probability based on the actual recorded results of repeated performances of an experiment.
• Key historical coin-tossing experiments demonstrating the empirical approach include:
  • Comte de Buffon ($1707$-$1788$): Tossed a coin $4040$ times and recorded $2048$ heads. The empirical probability of obtaining a head was calculated as:
    $\frac{2048}{4040} \approx 0.507$
  • John Edmund Kerrich ($1903$-$1985$): While interned in Denmark during World War II, tossed a coin $10000$ times and recorded $5067$ heads. The empirical probability of obtaining a head was calculated as:
    $\frac{5067}{10000} = 0.5067$
  • Karl Pearson ($1857$-$1936$): Tossed a coin $24000$ times and recorded $12012$ heads. The empirical probability of obtaining a head was calculated as:
    $\frac{12012}{24000} = 0.5005$
• These historical records illustrate the Law of Large Numbers: as the total number of experimental trials increases indefinitely, the experimental probability of an event approaches its theoretical probability of $0.5$.

Terminology and Fundamental Definitions

• Experiment: An operation or procedure that can produce one or several well-defined outcomes.
• Trial: An individual performance of an experiment. For example, flipping a coin once is a single trial of a coin-tossing experiment; rolling a six-sided die once is a single trial of a die-rolling experiment.
• Outcome: A specific possible result generated in a single trial of an experiment.
• Event: A collection of one or more outcomes of an experiment, conventionally denoted by uppercase letters such as $E$, $A$, or $B$.
  • An event is said to occur in a trial if the outcome of that trial satisfies the conditions defining the event.
  • Elementary Event: An event consisting of exactly one individual outcome.
  • Compound Event: An event consisting of two or more combined outcomes.

Mathematical Formulation of Empirical Probability

• Empirical probability, also designated as experimental probability, is calculated directly from data obtained through observation and experimentation.
• Let $n$ represent the total number of trials conducted in an experiment.
• Let $m$ represent the frequency with which a specific event $E$ occurs during those $n$ trials.
• The empirical probability of event $E$, denoted by $P(E)$, is defined as:
  $P(E) = \frac{\text{Number of trials in which event } E \text{ occurred}}{\text{Total number of trials conducted}}$
• In algebraic form:
  $P(E) = \frac{m}{n}$

Core Properties and Theorems of Probability

• Range of Probability Values:
  
  • The number of times an event occurs ($m$) can never be negative, and it cannot exceed the total number of trials ($n$):
    $0 \le m \le n$
  • Dividing the entire inequality by $n$ (where $n > 0$) yields the probability boundary theorem:
    $0 \le P(E) \le 1$
• Impossible Event:
  
  • If an event cannot possibly happen in any trial, the frequency of its occurrence is $m = 0$.
  • The empirical probability of an impossible event is identically zero:
    $P(E) = \frac{0}{n} = 0$
• Sure or Certain Event:
  
  • If an event is guaranteed to happen in every single trial, the frequency of its occurrence is $m = n$.
  • The empirical probability of a sure or certain event is identically one:
    $P(E) = \frac{n}{n} = 1$
• Sum of All Probabilities:
  
  • For an experiment comprising all possible mutually exclusive and exhaustive elementary events $E_1, E_2, \dots, E_k$, the sum of their individual empirical probabilities equals exactly $1$:
    $\sum_{i=1}^{k} P(E_i) = P(E_1) + P(E_2) + \dots + P(E_k) = 1$
• Complementary Events:
  
  • The non-occurrence of an event $E$ is termed the complement of $E$, denoted as $\overline{E}$ or $E'$ ("not $E$").
  • If an event occurs in $m$ out of $n$ trials, it fails to occur in $n - m$ trials.
  • The probability of the complementary event is:
    $P(\overline{E}) = \frac{n - m}{n} = 1 - \frac{m}{n} = 1 - P(E)$
  • Consequently:
    $P(E) + P(\overline{E}) = 1$

Standard Scenarios and Problem Archetypes

• Cricket Ball Frequency Model:

  • Scenario: A batswoman plays $30$ balls and hits a boundary $6$ times.
  • Total trials:
    $n = 30$
  • Number of boundary outcomes:
    $m = 6$
  • Number of non-boundary outcomes:
    $30 - 6 = 24$
  • Probability of hitting a boundary:
    $P(\text{Boundary}) = \frac{6}{30} = \frac{1}{5} = 0.2$
  • Probability of not hitting a boundary:
    $P(\text{No Boundary}) = \frac{24}{30} = \frac{4}{5} = 0.8$
  • Verification of complement:
    $P(\text{Boundary}) + P(\text{No Boundary}) = 0.2 + 0.8 = 1$

• Coin Toss Experiment (Two Coins):

  • Scenario: Two coins are tossed simultaneously $500$ times with the following frequency distribution:
  • Two heads ($E_2$): $105$ times
  • One head ($E_1$): $275$ times
  • No heads / zero heads ($E_0$): $120$ times
  • Total trials:
    $n = 500$
  • Individual probabilities:
    $P(E_2) = \frac{105}{500} = 0.21$
    $P(E_1) = \frac{275}{500} = 0.55$
    $P(E_0) = \frac{120}{500} = 0.24$
  • Summation check:
    $P(E_2) + P(E_1) + P(E_0) = 0.21 + 0.55 + 0.24 = 1.00$

• Die Rolling Experiment:

  • Scenario: A single die is rolled $1000$ times with recorded frequencies for outcomes $1$ through $6$:
  • Outcome $1$: $179$ times
  • Outcome $2$: $150$ times
  • Outcome $3$: $157$ times
  • Outcome $4$: $149$ times
  • Outcome $5$: $175$ times
  • Outcome $6$: $190$ times
  • Empirical probability of each face $i$:
    $P(1) = \frac{179}{1000} = 0.179$
    $P(2) = \frac{150}{1000} = 0.150$
    $P(3) = \frac{157}{1000} = 0.157$
    $P(4) = \frac{149}{1000} = 0.149$
    $P(5) = \frac{175}{1000} = 0.175$
    $P(6) = \frac{190}{1000} = 0.190$
  • Verification of total probability:
    $\sum_{i=1}^{6} P(i) = 0.179 + 0.150 + 0.157 + 0.149 + 0.175 + 0.190 = 1.000$

• Demographic Household Survey (Two Children per Family):

  • Scenario: $1500$ families with $2$ children were surveyed at random, yielding:
  • Families with $2$ girls: $475$
  • Families with $1$ girl: $814$
  • Families with $0$ girls: $211$
  • Total families ($n$):
    $n = 1500$
  • Probabilities:
    $P(\text{Family with } 2 \text{ girls}) = \frac{475}{1500} = \frac{19}{60}$$P(\text{Family with } 1 \text{ girl}) = \frac{814}{1500} = \frac{407}{750}$$P(\text{Family with } 0 \text{ girls}) = \frac{211}{1500}$
  • Summation:
    $\frac{475}{1500} + \frac{814}{1500} + \frac{211}{1500} = \frac{1500}{1500} = 1$

• Quality Control and Life of Tyres:

  • Scenario: A company tracks the distance travelled before a tyre requires replacement across $1000$ tyres:
  • Less than $4000\,km$: $20$ tyres
  • $4000\,km$ to $9000\,km$: $210$ tyres
  • $9001\,km$ to $14000\,km$: $325$ tyres
  • More than $14000\,km$: $445$ tyres
  • Probability that a tyre needs replacement before covering $4000\,km$:
    $P(< 4000\,km) = \frac{20}{1000} = 0.02$
  • Probability that a tyre lasts more than $9000\,km$ ($325 + 445 = 770$ tyres):
    $P(> 9000\,km) = \frac{770}{1000} = 0.77$

Comparison of Empirical and Theoretical Probability

• Empirical (Experimental) Probability:
  
  • Dependent on actual historical or experimental data.
  • Requires performing physical trials and recording frequencies.
  • May produce different results across different sets of trials of identical sizes.
  • Formula:
    $P(E) = \frac{\text{Frequency of event occurrence}}{\text{Total number of trials performed}}$
• Theoretical (Classical) Probability:
  
  • Assumes all elementary outcomes of an experiment are equally likely.
  • Does not require conducting physical experiments.
  • Yields a single deterministic value deduced from abstract mathematical symmetry.
  • Formula:
    $P(E) = \frac{\text{Number of outcomes favourable to } E}{\text{Total number of possible outcomes}}$
• Connection and Convergence:
  
  • Empirical probability provides an estimate of theoretical probability.
  • As the number of trials $n \rightarrow \infty$, the empirical probability approaches the theoretical probability asymptotically.