### Random variables and Probability distribution of a random variable (Probability 9 NCERT)

Random variables We have already learnt about random experiments and formation of sample spaces. In most of these experiments, we were not only interested in the particular outcome that occurs but rather in some number associated with that outcomes as shown in following examples/experiments. (i) In tossing two dice, we may be interested in the […]

### Bayes’ theorem (Probability 8 NCERT)

Bayes’ theorem Consider that there are two bags I and II. Bag I contains 2 white and 3 red balls and Bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags. We can find the probability of selecting any of the bags (i.e. 1 /2 […]

### Partition of a sample space and Theorem of total probability (Probability 7 NCERT)

Partition of a sample space A set of events E1, E2, …, En is said to represent a partition of the sample space S if  In other words, the events E1, E2, …, En represent a partition of the sample space S if they are pairwise disjoint, exhaustive and have nonzero probabilities.  As an example, we see that any […]

### Independent events (Probability 6 NCERT)

Independent events Consider the experiment of drawing a card from a deck of 52 playing cards, in which the elementary events are assumed to be equally likely. If E and F denote the events ‘the card drawn is a spade’ and ‘the card drawn is an ace’ respectively, then  Also E and F is the […]

### Multiplication rule of probability

Multiplication rule of probability Let E and F be two events associated with a sample space S. Clearly, the set E ∩ F denotes the event that both E and F have occurred. In other words, E ∩ F denotes the simultaneous occurrence of the events E and F. The event E ∩ F is […]

### Conditional probability(Probability 4 NCERT)

Conditional probability Uptill now in probability, we have discussed the methods of finding the probability of events we will see now conditional probability of an event. If we have two events from the same sample space, does the information about the occurrence of one of the events affect the probability of the other event? Let […]

### Axiomatic approach to probability ( Probability 3 NCERT)

Axiomatic approach to probability In earlier topics, we have considered random experiments, sample space and events associated with these experiments. In our day to day life we use many words about the chances of occurrence of events. Probability theory attempts to quantify these chances of occurrence or non occurrence of events.  Axiomatic approach is another way […]

### Event in probability ( Probability 2 NCERT)

Event in probability We have studied about random experiment and sample space associated with an experiment. The sample space serves as an universal set for all questions concerned with the experiment.  Consider the experiment of tossing a coin two times. An associated sample space is  S = {HH, HT, TH, TT}.  Now suppose that we […]

### Random experiments outcome and sample space ( Probability 1 NCERT)

Random experiments, outcome and sample space In our day to day life, we perform many activities which have a fixed result no matter any number of times they are repeated. For example given any triangle, without knowing the three angles, we can definitely say that the sum of measure of angles is 180°.  We also […]

### Polar representation of a complex number(Complex Number 4 NCERT)

Polar representation of a complex number Let the point P represent the nonzero complex number z = x + iy. Let the directed line segment OP be of length r and θ be the angle which OP makes with the positive direction of x-axis (Fig 5.4). We may note that the point P is uniquely […]