Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM
http://fb.me/thehgclasses hgoyalclasses@gmail.com 9599697178.
Hg
PROBABILITY
• Conditional Probability: The conditional probability of an event E, given the occurrence of the event F is denoted
as P(E | F) read as Probability of E given F and is given by:
o P(E | F) = P(E ∩ F) / P(F); P(F) ≠ 0.
o 0 ≤ P(E|F) ≤ 1
o P(E’ | F) = 1 - P(E | F)
o P((E U F) |G) = P (E | G) + P(F | G) – P((E ∩ F) | G)
• Multiplication Theorem on Probability: If E and F be two events associated with a Sample Space S. E ∩ F or EF
denotes the event that both E and F has occurred.
o P(E ∩ F) or P(EF) = P(E | F) x P(F)
For more than 2 events, E, F and G:
o P(E ∩ F ∩ G) = P (E | F ∩ G) x P(F ∩ G)
= P (E | F ∩ G) x P (F | G) x P(G)
Or P(EFG) = P(E | FG) x P(F | G) x P(G)
• Independent Events: Two events E and F are said to be independent if probability of occurrence of E isn’t
affected by the probability of occurrence of the other and vice-versa.
In other words, for independent events:
o P(E | F) = P(E)
o P(F | E) = P(F)
o P(E ∩ F) = P(E) x P(F) and P(E ∩ F ∩ G) = P(E) x P(F) x P(G) [by multiplication rule]
• Total Probability Theorem: Let E
1
, E
2
, E
3
… … … E
n
are the events which constitute a partition of Sample Space S,
i.e. each of E
1
, E
2
, E
3
… … E
n
are pair wise disjoint and E
1
U E
2
U E
3
U … … U E
n
= S and each of P(E
i
) ≠ 0; then the
probability of event A associated with S is given by:
P(A) = P(A ∩ E
1
) + P(A ∩ E
2
) + … … … + P(A ∩ E
n
)
= P(A | E
1
) x P(E
1
) + P(A | E
2
) x P(E
2
) + … … … + P(A | E
n
) x P(E
n
)
OR P(A) =
• Bayes’ Theorem: P(E
k
| A) =
for any k = 1, 2, 3 … n
• Random Variable: A random variable is a real valued function whose domain is the sample space of a random
experiment. The values a random variable takes directly depend on the outcomes in the sample space.
e.g. In the random experiment of the throw of two dies, the random variable X is defined as the sum of the
numbers appearing. In this case, X can take any natural number in the interval [2, 12].
• Probability Distribution of a random variable: It is the following system of numbers:
where, p
i
> 0,
= 1
X
x
1
x
2
…
x
n
P(X)
p
1
p
2
…
p
n
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
___________________________________________________________________________________________________________________________________
___________________________________________________________________________________________________________________________________
The Hg Classes (8
th
to 12
th
) By: Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM
http://fb.me/thehgclasses hgoyalclasses@gmail.com 9599697178.
Hg
• Mean or Expected Value of X ():
o E(X) =
• Variance of X (
):
o Var(X) = E(X - )
2
o Var(X) =
o Var(X) = E(X
2
) – [E(X)]
2
o S.D. (X) =
=
• Bernoulli Trials: Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions:
o They should be finite
o They should be independent
o Each trial as exactly two outcomes: success or failure
o The probability of success (p) remains the same in each trial. (probability of failure becomes = 1-p).
Examples:
o Throwing a die 50 times with p = probability of getting an even number = 1/2.
o Throwing a coin 100 times with p = probability of getting a head = 1/2.
• Binomial Distribution: For n-Bernoulli trials, the number of successes obtained becomes a random variable which
can take a value from X = 0 to X = n; and its probability distribution is a binomial distribution denoted by B(n, p)
where p is the probability of success.
The probability of getting x successes in n-Bernoulli trials
= P(x) = Probability function of the binomial distribution
=
p
x
(1-p)
(n-x)
Or
p
x
q
(n-x)
, where q = 1-p = probability of failure.
Thus, the binomial distribution B(n, p) will be as given below:
