Comprehensive theory, key formulas, diagrams, and memory aids for Probability.
In everyday conversation, we often use words like "probably," "chance," "doubt," and "likely." For instance, "It will probably rain today," or "There is a high chance India will win the match." These words introduce an element of uncertainty. Mathematics attempts to measure this uncertainty using a concept called Probability.
In this chapter, we will look at the experimental (or empirical) approach to probability, which is based on actual observations and experiments.
Let's say you toss a coin. You know it can either land on Heads or Tails. If you toss it $10$ times, you might get $6$ Heads and $4$ Tails. If you toss it $100$ times, you might get $53$ Heads and $47$ Tails.
The empirical (or experimental) probability $P(E)$ of an event $E$ happening is defined as:
$$P(E) = \frac{\text{Number of trials in which the event happened}}{\text{Total number of trials}}$$
A die is thrown $1000$ times with the frequencies for the outcomes $1, 2, 3, 4, 5,$ and $6$ as given in the following table:
| Outcome | Frequency |
|---|---|
| 1 | 179 |
| 2 | 150 |
| 3 | 157 |
| 4 | 149 |
| 5 | 175 |
| 6 | 190 |
What is the probability of getting a 3? * Total number of trials = $1000$ * Number of trials where the outcome was $3$ = $157$ * $P(getting a 3) = \frac{157}{1000} = 0.157$
By looking at the formula for empirical probability, we can deduce a few strict mathematical rules that probability must always follow.
The number of times an event happens can never be less than $0$, and it can never be more than the total number of trials. Therefore, the numerator is always between $0$ and the denominator. Thus, the probability of an event $E$ always lies between $0$ and $1$ (inclusive). $0 \le P(E) \le 1$
If you calculate the probability of every possible outcome of an experiment and add them together, the sum will always be exactly $1$. Let's look at the die example again: $P(1) = 0.179$ $P(2) = 0.150$ $P(3) = 0.157$ $P(4) = 0.149$ $P(5) = 0.175$ $P(6) = 0.190$
Sum = $0.179 + 0.150 + 0.157 + 0.149 + 0.175 + 0.190 = 1.000$
In Grade 9, the focus is on experimental probability, which depends entirely on the data gathered from actually performing the experiment. If you toss a coin $10$ times, your experimental probability of getting Heads might be $0.7$.
In higher classes, you will study theoretical (or classical) probability, which predicts what should happen in an ideal world without needing to do the experiment. Theoretically, the probability of getting Heads is exactly $0.5$.
An interesting fact about probability is the Law of Large Numbers: As you increase the number of trials (e.g., tossing a coin $10,000$ times instead of $10$), the experimental probability gets closer and closer to the theoretical probability.
Probability is not just for games of chance. It forms the foundation of many modern fields: * Meteorology: Weather forecasting uses complex models to give a percentage chance of rain. * Insurance: Actuaries use probability to determine life expectancy and calculate insurance premiums. * Medical Science: Evaluating the success rate and potential side effects of new drugs. * Quality Control: Factories test a small sample of products to determine the probability that the entire batch is defect-free.
Probability is the mathematical tool for quantifying uncertainty. In the empirical approach, we rely on historical data and actual trials to predict the likelihood of future events. Remembering that probability is always a fraction between $0$ and $1$, and that all possible probabilities sum to $1$, helps keep calculations grounded in reality.