**Probability - Class 10 : ****Notes**

**(1) In the experimental approach to probability, We find the probability of the occurrence of event by actually performing the experiment a number of times and adequate recording of the happening of event.**

**(2) In the theoretical approach to probability, we try to predict what will happen without actually performing the experiment.**

**(3) An outcome of a random experiment is called an elementary event.**

* For Example: *Consider the random experiment of tossing coin. The possible outcome of this experiment are head(H) and tail (T). if we define E1 = getting head(H), E2 = getting tail (T)

Then, E1 and E2 are elementary associated with the experiments of tossing of a coin.

**(4) An event associated to a random experiment is a compound event if it is obtained by combining two or more elementary events associated to the random experiment.**

* For Example: *In a single throw of a die, the event “getting an even number” is a compound event as it is obtained by combining three elementary events, namely 2, 4, 6.

**(5) An event A associated to a random experiment is said to occur if any one of the elementary events associated to the event A is an outcome.**

* For Example: *Consider the random experiment of throwing an unbiased die. Let A denote the event “getting an even number ”. Elementary events associated to this event are 2, 4, 6. Now, suppose that in a trail the outcome is 4, then we say that the event A has occurred. In another trail , let the outcome be 3, then we say that the event A has not occurred.

**(6) An elementary event is said to be favourable to a compound event A, if it satisfies the definition of the compound event.**

* For Example: *Consider the random experiment of two coins are tossed simultaneously and A is an event associated to it defined as “getting exactly one head”. We say that the event A occurs if we get either HT or TH as an outcome. So, there are two elementary events favourable to the event A.

**(7) If there are n elementary events associated with a random experiment and m of them are favourable to an event A, then the probability of happening occurrence of event A is denoted by P(A)and is defined as the ratio mn**

**i.e., P(A)=mn**

* For Example: *Let A denote the event “getting an even number”

Clearly , event A occurs if we obtain any one of 2, 4, 6 as an outcome.

Favourable number of elementary events=3

P(A)=36=12

**(8) For any event A associated to a random experiment, we have**

**(i) 0≤P(A)≤1 (ii) P(A)¯¯¯¯¯¯¯¯¯¯¯=1−P(A)**

**Proof of (i):**

P(A)=mn

0≤m<n

0≤mn≤1

0≤P(A)≤1

**Proof of (ii):**

If P(A)=1, then A is called a certain event and A is called an impossible event, if P(A)=0.

If m elementary events are favourable to an event A out of n elementary events, then the number of elementary events which ensure the non-occurrence of A. i.e. the occurrence of A¯ is n−m

P(A¯¯¯¯)=n−mn

P(A¯¯¯¯)=1−mn

P(A¯¯¯¯)=1−P(A)

**(9) The probability of a sure event is 1.**

* For Example: *sun is rising from the east. this is a sure event. so probability of sure event is 1

**(10) The probability of an impossible event is 0.**

* For Example: *suppose the sun is rising from the west. this event is impossible event so probability of impossible event is always 0.

**(11) The sum of the probabilities of all the outcomes (elementary events) of an experiment is 1.**

* For Example: *Suppose in experiment of tossing coin 10 times 6 time head appear and 4 times tails appear.

So probability of getting head is P(H)=mn

where m is number of time head appear And n is number of time tossing coin.

So, P(H)=610=0.6.

Now probability of getting tail is given by P(T)=mn

where m is number of time tail appear and n is number of time tossing coin.

So, P(T)=410=0.4.

So total probability of this experiment is given by P(A)=P(H)+P(T). P(A)=0.6+0.4 P(A)=1.

Hence sum of probability of all outcomes of an experiment is always 1.