Notes on Probability- MATHS (CBSE) 2022

PROBABILITY

🔻The sum of all the probabilities in the sample space is 1.

🔻The probability of an event which cannot occur is 0.

🔻The probability of any event which is not in the sample space is zero.

🔻The probability of an event which must occur is 1.

  • The probability of the sample space is 1.
  • The probability of an event not occurring is one minus the probability of it occurring.

🔻The complement of an event E is denoted as E’ and is written as P (E’) = 1 – P (E)

P (A∪B) is written as P (A + B) and P (A ∩ B) is written as P (AB).

🔻If A and B are mutually exclusive events, P(A or B) = P (A) + P (B)

🔻When two events A and B are independent i.e. when event A has no effect on the probability of event B, the conditional probability of event B given event A is simply the probability of event B, that is P(B).

If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by P (A and B) = P (A) P (B|A).

🔻A and B are independent if P (B/A) = P(B) and P(A/B) = P(A).

🔻If E1, E2, ……… En are n independent events then P (E1 ∩ E2 ∩ … ∩ En) = P (E1) P (E2) P (E3)…P (En).

🔻Events E1, E2, E3, ……… En will be pairwise independent if P(Ai ∩ Aj) = P(Ai) P(Aj) i ≠ j.

P(Hi | A) = P(A | Hi) P(Hi) / ∑i P(A | Hi) P(Hi).

🔻If A1, A2, ……An are exhaustive events and S is the sample space, then A1 U A2 U A3 U …………… U An = S

🔻If E1, E2,….., En are mutually exclusive events, then P(E1 U E2 U …… U En) = ∑P(Ei)

🔻If the events are not mutually exclusive then P (A or B) = P (A) +P (B) – P (A and B)

🔻Three events A, B and C are said to be mutually independent if P(A∩B) = P(A).P(B), P(B∩C) = P(B).P(C), P(A∩C) = P(A).P(C), P(A∩B∩C) = P(A).P(B).P(C)

The concept of mutually exclusive events is set theoretic in nature while the concept of independent events is probabilistic in nature.

🔻If two events A and B are mutually exclusive,

P (A ∩ B) = 0 but P(A) P(B) ≠ 0 (In general)

🔻P(A ∩ B) ≠ P(A) P(B)

🔻 Mutually exclusive events will not be independent.

The probability distribution of a count variable X is said to be the binomial distribution with parameters n and abbreviated B (n,p) if it satisfies the following conditions:

  • The total number of observations is fixed
  • The observations are independent.
  • Each outcome represents either a success or a failure.
  • The probability of success i.e. p is same for every outcome.

🔻Some important facts related to binomial distribution:

  • (p + q)n = C0Pn + C1Pn-1q +…… Crpn-rqr +…+ Cnqn
  • The probability of getting at least k successes out of n trials is P(x > k) = Σnx = k nCxpxqn-x Σnx = k nCxqn-xpx = (q + p)n = 1
  • Mean of binomial distribution is np
  • Variance is npq

Standard deviation is given by (npq)1/2, where n Sum of binomials is also binomial i.e. if X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable with distribution X + Y ~ B(n + m, p). If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distributionY ~ B( n, pq). The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B (1, p) has the same meaning as X ~ Bern (p).If an experiment has only two possible outcomes, then it is said to be a Bernoulli trial. The two outcomes are success and failure.Any binomial distribution, B (n, p), is the distribution of the sum of n independent Bernoulli trials Bern (p), each with the same probability p.

The binomial distribution is a special case of the Poisson Binomial Distribution which is a sum of n independent non-identical Bernoulli trials Bern(pi). If X has the Poisson binomial distribution with p1 = … = pn = p then X ~ B(n, p).A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).

 

 

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