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PROBABILITY FOR CLASS X
Let's understand some basic terms for this chapter.

* Random Experiment

: Any action which can give one or more results is called a random experiment
or trial is called random experiment

* Sample space

: The set of all possible outcomes of an experiment or trial is
called the sample space of that experiment n is denoted by S.
1.Consider the simple experiment of tossing a coin.Since the coin can either heads or tails, the sample space S would be
S={h,t} i.e 2

Event

:Any action occured is called event.

Independet event

: Two events E1 and E2 are said to be independent if the
occurence of one doesn't depend on the occurence of other
of getting Two event of getting a head on the first coin.
Example:Suppose we toss two coins.
Let E1=event of getting a head on the first coin
and E2=Event of getting a head on the second coin
Event E1 and E2 are called independent event

Sure event :

An event which always happens is called a sure event.

Impossible event:

An event which never happens is called impossible event.

Elementry event :

An elementry event which has one (favourable)outcome .
from the sample space is called elementry event.

Compound event:

An event which has more than one(favourable)
outcome from the sample space is called Compound event.

Complementary event :

If E is an event,then the event 'not E' is complementary event of E.
If S is the finite sample space of an experiment and every outcome of S
is equally likely and if S is is equally likely and if E is an event (i.e,E,S)=n(E)/n(s), n(E)= no.of events and n(s)= no.of space.
For two indepedent events E1 and E2,we have
i) P(E1compE2)=p(E1)*p(E2),comp meaning is compliment
ii) p(E1compE2')=p(E1)*p(E2'), E2' = not E2
iii) p(E1'comp E2)=p(E1')*p(E2),E1'= not E1
iv) p(E1'comp E2')= p(E1')*p(E2')

* * * * * * * CHECK OUT SOME SOLVED EXAMPLES* * * * * * *

Example1:A coin is tossed 1000 times and the outcomes are noted as 1000 times
and the outcomes are noted as:
HEAD : 687, TAIL:313,Find the Probability of the coin coming up with i) a head ii) a tail
Ans: i)p(E1)=687/1000

ii) p(E2)=313/1000
Check the answer
EXAMPLE 2.A dice is thrown 700 times and the frequencies of the outcomes
1,2,3,4,5,6 were recorded as given below:
Outcomes: 1, 2, 3, 4, 5, 6
Frequency : 198,89, 99,96,120,98
Find the probability of getting
i) each outcome ii) an even no iii)a number less than 3
Ans:i) For 1,P(E1) = 198/700, for 2 p(E2)= 89/700
for 3, p(E3)= 99/700, for 4 p(E4) = 96/700,p(E5)=98/700 ,

ii) 2,4,6 for 2,p(Ee) = (89+96+98)/700 = 283/700

iii) p(E1)=198/700,p(E2)=89/700
so, probability less than 3=p(E1)+p(E2)=(198 +89)/700 = 287/700
Check the answer
Example:3.Let E1 and E2 be two events such that p(E1)=4/7 and p(E2)=1/4
i)P(E1 or E2) ,if E1 and E2 are mutually exclusive events
ii)p(E1 and E2),if E1 and E2 are indepedent events
Ans : i) As E1 and E2 be two mutually exclusive events then,
E1compE2 = null

p(E1 or E2)=p(E1 U E2)
=p(E1)+p(E2)=(4/7)+(1/4)=23/28

ii) P(E1 comp E2)
=p(E1)*P(E2)= (4/7)*(1/4)= 1/7

Check the answer
Example:4.Let E1 and E2 be events such that p(E1)=0.3,p(E1 U E2)=0.4,
Find the value of p(E2).
Ans: p(E1 or E2)=p(E1 U E2)
=p(E1)+p(E2)=(0.3)+p(E2)=0.4
P(E2)= 0.4-0.3=0.1
Check the answer
Example:5. If E1 and E2 are two independent events
such that P(E1) =0.35 and P(E1 U E2)=0.60,find p(E2).
Ans: Let p(E2)=x.
Then,E1 n E2 being independent events,we
p(E1 comp E2) = p(E1)*P(E2)=0.35*p(E2)
p(E1 U E2)=0.60
p(E1)+p(E2)-p(E1)*p(E1)
0.60=p(E1)+p(E2)-0.35*p(E2)
0.60=0.35+(1-0.35)*p(E2)
0.25=0.65*p(E2)
p(E2)=0.65/0.25=13/5=2.6
Check the answer
Example 6: An unbiased dice is tossed twice.
Find the probability of getting a 4,5 or 6.
on the first toss and a 1,2,3 or 4 on the second toss.
Ans:Sample space S= { 1,2,3,4,5,6}
E1={4,5,6} on first toss.
E2= {1,2,3,4} on second toss.
p(E1) =3/6 = 1/2
p(E2)= 4/6 = 2/3
As it is independent event
probability=p(E1 comp E2)= (1/2)*(2/3) = 1/3
Check the answer
Example 7.A speaks the truth in 60% of the cases, and B in the 90% of the cases.
In what percentage of cases are they likely to contradict each other in stating the same fact?.
Ans: it means at least one person is telling lie
p(E1)=6/10=3/5,p(not E1) = 1-(3/5)=(5-3)/5 = 2/5
p(E2) = 9/10,p(not E2)=1-9/10=1/10
case I: When both are telling lie.
p(not E1 and not E2) = (2/5)*(1/10) = 1/25
p(not E1 and E2) = (2/5)*(9/10) = 9/25
p(not E2 and E1) = (3/5)*(1/10 ) = 3/50
(1/25) +(9/25)+(3/50)=(23/50) *100 = 46%
Check the answer
Example 8.A problem is given to three students whose chance of solving it are
1/3,2/7 and 3/8.What is the probability that the problem will be solved.
Ans:Let the three students be named A,B,C respectively.
E1=1/3,E2=2/7, E3=3/8
p((not E1) comp(not E2) comp(not E3)) = (2/3)*(5/7)*(5/8)
= 50/168 = 25/84 = (1-25/84)= (84-25)/84 = 59/84
Check the answer

**********SOLVE THE FOLLOWING PROBLEM(EASY LEVEL)*******

1.Suresh and Ramesh play loodo which has four colors red,white,yellow and green.

Find the probability i)Suresh choose yellow color ii)Ramesh choose green color.

2. If one dice is thrown what be the probability that the number would be less than 3.

3. Two dice are thrown simultonusly what would be the probability that both dice no.have even no.

4. Five coines are tossed together find the probability that i) either all coins are head or tail. ii) neither all coins are head nor tail.

5. 52 cards are shuffled find the probability that cards are either red or black.

6. The ratio of boys and girls are 5:8 ,What is the probability that students are boys.

7. 20% students play football,25% play hockey,30% play soccer rest students don't play any outdoor games. Find the probability.
that i) no students play football ii)either students play soccer or hockey iii) students play both football and hockey.

8.3 red balls,4 green balls and 5 yellow balls are kept in a bucket.What is the probability that

i) three balls taken at a time would be red. ii) five balls taken at a time would be 2 red balls,2 green balls and 1 yellow ball.

9.One teen goes to bookshop where he gets the collection of 57 fiction books, 67 non-fiction and 83 religious books.What is the probability that
that i) teen choose fiction books ii) teen doesn't choose non-fiction books.

10.Two dice are thrown simultonusly, What is the probability that
i) both dice have six. ii) both dice have even no. iii) Either of dice has odd no.

************* SOLVE THE FOLLOWING PROBLEM (HARD LEVEL)**********

 1.If A and B are two independent events such that p(A)= 1/3 and p(B)= 3/5, find I
      i) p(A and B) ii)p(A or B) iii)p(not A and B) iv)p(A not B) v) p( not A and not B) 

     2.If A and B are two events such that p(A)=1/4,p(B)=1/3, and p(A U B)=1/2,
     show that A and B are independent events.

     3.Let A and B be events such that p(A)=1/2,p(B)=7/12 and p(A U B) = 1/2,show that they are independent events or not.

     4.Let A and B be events such that p(A)=1/2,p(B)=7/12 and p(not A or not B)=1/4 .Check whether they are independent or not.

     5.A can hit a target 4 times in 5 shots,B can  hit 3 times in 4 shots, and C can hit 2 times in 3 shots.Calculate the probability
       that i)A,B and C all hit the target ii)B,C and A doesn't hit the target.

    6.Kamal and Vimal appeared for an interview for two  vacancies. The probability of  Kamal's selection is 1/3  and that of Vimal's
      selection  is 1/5.Find the probability  that only one of them is selected.

   7.In a school 30% plays football,20% plays hockey,20% plays badmintion, What is the probability that students play all  three 
     outdoor games.
   
   8.In a university of Science students (2/3)  study Physics,(4/5) study Chemistry and  (5/6) study Maths . What is the probabillity
     students study either of two subjects.

  9.A doctor claims  that  60%  of  10 patients checked is allergic to dust  and 30% is allergic to flower.Find the probability of
    first four be allergic.

10. A and B appear for an interview of two vacancies in the same post. The probability of A's  selection is 1/6 and
    that of B's selection is 1/4.Find the probability  that
    i) Both of them are selected
    ii) Neither of them are selected
    iii)at least one of them is selected
    iv) only one of them is selected

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