CBSE Class 9 Mathematics Probability Notes Set 12

Experiment, Trial and Event

A trial is one performance of an experiment, such as tossing a coin or throwing a die. Each possible result is an outcome, and an event is one or more outcomes we are interested in. The set of all possible outcomes is called the sample space. For one die it is 1, 2, 3, 4, 5, 6 and for two coins it is HH, HT, TH, TT.

Experimental (Empirical) Probability

If an experiment is repeated n times and an event A occurs in m of those trials, the empirical probability of A is P(A) = m / n, that is, the number of trials in which A happens divided by the total number of trials.
Since m lies between 0 and n, every probability lies between 0 and 1. An event with probability 1 is certain and an event with probability 0 is impossible.

Complementary Events

If A is an event and 'not A' is the event that A does not happen, then P(A) + P(not A) = 1, so P(not A) = 1 - P(A). This is useful for 'at least one' questions: find the probability of none and subtract from 1.
Example: if the probability of winning is 0.3, then the probability of losing is 0.7.

Theoretical Probability

When all outcomes are equally likely, P(A) = number of outcomes favourable to A divided by the total number of outcomes. We can find it without performing the experiment.
For two coins the sample space has 4 outcomes, so P(two heads) = 1/4 and P(at least one head) = 3/4. For three coins there are 8 outcomes, and P(exactly two heads) = 3/8.

Playing Cards and Dice Facts

A pack has 52 cards: 26 red and 26 black, in four suits (hearts, diamonds, spades, clubs) of 13 cards each. There are 12 court cards (kings, queens, jacks) and 16 honour cards (the 12 court cards and 4 aces).
A die has six faces numbered 1 to 6; the primes among them are 2, 3 and 5.

Types of Experiment

In a deterministic experiment the result is the same every time under identical conditions. In a random (probabilistic) experiment the result may differ each time and is one of several possible outcomes. Probability is studied for random experiments.

Solving Frequency Table Questions

Read the table, find the total number of trials, count the trials favourable to the event, and divide. For 'at least' include the stated value and everything above it; for 'at most' include the stated value and everything below it; for 'more than' do not include the stated value. Check that the frequencies add up to the total given in the question.