1. Mathematical definition of probability:
Probability of an event
Note:
(i) 0 £ p (A) £ 1
(ii) probability of an impossible event is zero
(iii) probability of a sure event is one
(iv) P(A) + P(Not A) = 1 i.e. P(A) + P(`A) = 1
2. Odds for an event:
If P(A) = (m/n) and P(`A) = ((n-m) / n)
Then odds in favour of A = P(A)/P(`A) = m / (n-m)
And adds in against of A P(A) / P(`A) = (n-m) / m
3. Set theoretical notation of probability and some important result:
(i) P(A+B) = 1 – P(`A`B)
(ii) P(A/B) = P(AB) / P(B)
(iii) P(A+B) = P(AB) + P(`AB) + P(A`B)
(iv) A Ì B Þ P(A) £ P(B)
(v) P(`AB) = P(B) – P(AB)
(vi) P(AB) £ P(A) P(B) £P(A+B) £ £ P(A) + P(B)
(vii) P(Exactly one event ) = P(A`B) + P(`AB)
(viii) P(`A+`B)= 1- P(AB) = P(A) + P(B) – 2P(AB)
(ix) P(neither A nor B) = P(`A`B) = 1 – P(A+B)
(x) when a coin is tossed n times or n coin are tossed once, the probabilty of each simple event is (1/2n)
(xi) when a dice is rolled n times or n dice are rolled once, the probability of each simple event is (1/66.
(xii) when n cards are drawn (1£ n 52) form well shuffled deck of 52 card, the probability of each simple event is (1/52Cn).
(xiii) If n cards are drawn one after the other with replacement, the probability of each simple event is 1/(52)n
(xiv) P(none) = 1 –P (atleast one)
(xv) Playing card:
(a) Total card: 52 (26 red, 26 black)
(b) four suits: heart, diamond, spade, club (13 cared each)
(c) court (face) cards : 12 (4 kings ,4 queens 4 jacks )
(d) Honour card: 16(4 Aces, 4 kings, 4 queens, 4 jacks )
(xvi) Probability regarding n letters and their envelopes: if n letters corresponding to n envelopes at random, then
(a) probability that all the letters are in right envelopes = (1/n!)
(b) probability that all letters are not in right envelopes = 1 – (1/n!)
(c) probability that no letter are in envelope
(d) probability that exactly r letters are in right
4. Addition Theorem of Probability:
(i) when events are mutually exclusive i.e. n (A Ç B)=0 Þ P(AÇB)=0
\ P(AÈB) = P(A) + P(B)
(ii) when events are not mutually exclusive i.e. P(AÇB) ¹ 0
\ P(AÈB) = P(A) + P(B) – P(AÇB)
or P(A+B) = P(A) + P(B) – P(AB)
(iii) when events are independent i.e. P(AÇB)= P(A) P(B)
\ P(A+B) = P(A) +P(B) –P(A) or P(B)
5. Conditional probability:
P(A/B) = probability of occurrence of A, given that B has alread happened = P(AÇB) / P(B) P(B/A) = probability of occurrence of B, given that A has already happened = P(AÇB) / P(A)
Note: If the outcomes of the experiment are equally
(i) If and B are independent event, then P(A/B) = P(A) and P(B/A) = P(B)
(ii) Multiplication Theorem:
P(AÇB) = P(A/B).P(B), P(B) ¹ 0
Or P(AÇB) = P(B/A)P(A),P(A) ¹ 0
Generalized:
P(E1ÇE2ÇE3 Ç ….. Ç En)
= P(E1)P(E2/E1) P(E3/E1 Ç E2) P(E4/E1 Ç E2 Ç E3 ) ….. If events are independent, then P(E1 Ç E2 Ç E3 Ç …… Ç En ) = P(E1) P(E2) ….. P(En)
6. Probability of at least at least one of the n Independent event:
If
P1, P2, ….. Pn are the probability of n independent events A, A2, ….. An then the probability of happening of at least one these event is .
1-[(1-P1)(1-P2) ….. (1-Pn)]
Or P(A1+A2 + ….. + An ) = 1 – P(`A1) P(`A2) ….. P(`An)
7. Total Probability:
Let A1, A2, ….. An are n mutually exdusive & set of exhaustive events and events A can occur through any one of these events , then probability of occurece of A P(A) = P(AÇA1) + P(AÇA2)+ …. + P(AÇAn)
8. Baye’s Rule:
Let A1,A2,A3 be any three mutually exclusive & exhaustive events (i.e. A1 È A2 ÈA3 = sample space & A1ÇA2A3 = Æ) an sample space S and B is any other event on sample space then,
9. Probability distribution:
(i) If random variable x assumed values x1, x2, …. Xn with probability P1, P2, ….. Pn respectively then
(a) P1 + P2 + P3 + ….. _ Pn =1
(b) mean E(x) = åPixi
(c) Variance = åx2Pi - (mean)2 = å(x2)- (E(x))2
(ii) Binomial distribution: If an experiment is repeated n times, the successive trials being independent of one another, then the probability of r success is nCrPrqn-r
Atleast r success is ån nCk Pk qn-k where p is probability of success in a single trial, q =1 –p
(a) mean E(x) = np
(b) E(x2) – npq + n2p2
(c) variance E(x2) – (E(x))2 = npq
(d) Standard deviation = Önpq
10. Truth of the statement:
(i) If two persons A and B speaks truth with the probability P1 & P2 respectively and if they agree on a statement, then the probability that they are speaking truth will be given by
(ii) If A and B both assert that an event has occurred, probability of occurrence of which is a then the probability that event has occurred. Given that the probability of A & B speaking truth is P1, P2.
(iii) If in the second part the probability that their lies (jhuth) coincides is b then form above case requird probability will be
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