- Classical Definition of Probability
- Type of Event - Impossible and Sure Or Certain
- assume that all the experiments have equally likely outcomes, impossible event, sure event or a certain event, complementary events,

We know, in advance, that the coin can only land in one of two possible ways - either head up or tail up (we dismiss the possibility of its ‘landing’ on its edge, which may be possible, for example, if it falls on sand). We can reasonably assume that each outcome, head or tail, is as likely to occur as the other. We refer to this by saying that the outcomes head and tail, are equally likely.

Suppose that a bag contains 4 red balls and 1 blue ball, and you draw a ball without looking into the bag. What are the outcomes? Are the outcomes — a red ball and a blue ball equally likely? Since there are 4 red balls and only one blue ball, you would agree that you are more likely to get a red ball than a blue ball. So, the outcomes (a red ball or a blue ball) are not equally likely. However, the outcome of drawing a ball of any colour from the bag is equally likely. So, all experiments do not necessarily have equally likely outcomes. However, in this chapter, from now on, we will assume that all the experiments have equally likely outcomes. In Class IX, we defined the experimental or empirical probability P(E) of an event E as

`P(E)="Number of trials in which the event happened"/"Total number of trials"`

In experiments where we are prepared to make certain assumptions, the repetition of an experiment can be avoided, as the assumptions help in directly calculating the exact (theoretical) probability. The assumption of equally likely outcomes (which is valid in many experiments, as in the two examples above, of a coin and of a die) is one such assumption that leads us to the following definition of probability of an event. The **theoretical probability** (also called classical probability) of an event E, written as P(E), is defined as

`P(E)="Number of outcomes favourable to E"/"Number of all possible outcomes of the experiment"`

**Example** 1 : Find the probability of getting a head when a coin is tossed once. Also find the probability of getting a tail.

**Solution** : In the experiment of tossing a coin once, the number of possible outcomes is two — Head (H) and Tail (T). Let E be the event ‘getting a head’. The number of outcomes favourable to E, (i.e., of getting a head) is 1. Therefore,

`P(E)=P (head)="Number of outcomes favourable to E"/"Number of all possible outcomes"=1/2`

Similarly, if F is the event ‘getting a tail’, then

`P(F)=P(tail)=1/2`