Probability
Class 12 Mathematics · Mathematics Part II · Chapter 7
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Mathematics
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Mathematics
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History
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English Core
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Chemistry
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हिंदी (आधार)
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Accountancy
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اردو (اختیاری)
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हिंदी (आधार)
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संस्कृतम् (केन्द्रिकम्)
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Physics
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हिंदी (ऐच्छिक)
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संस्कृत (ऐच्छिक)
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Physics
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हिंदी (ऐच्छिक)
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اردو (کور)
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Chemistry
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Urdu Elective
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Computer Science
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Chemistry
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Fine Art · Painting
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हिंदुस्तानी संगीत (गायन / वादन)
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Biology
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Informatics Practices
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Accountancy
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Biotechnology
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Accountancy
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सृजनात्मक लेखन और अनुवाद
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Business Studies
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हिंदुस्तानी संगीत (तबला / पखावज)
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Business Studies
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Economics
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Economics
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History
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History
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History
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Geography
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Geography
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Geography
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English Elective · Kaleidoscope
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Political Science
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Political Science
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Sociology
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Sociology
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Psychology
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English Core · Flamingo
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English Core · Vistas
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Home Science
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Home Science
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Summary
This chapter develops probability beyond simple counting. Conditional probability \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\) measures the chance of \(A\) given that \(B\) has occurred. The multiplication theorem follows: \(P(A\cap B)=P(B)\,P(A\mid B)\). Two events are independent when \(P(A\cap B)=P(A)\,P(B)\). The Theorem of Total Probability sums the contributions of mutually exclusive, exhaustive events, and Bayes' Theorem reverses conditioning to find \(P(E_i\mid A)\) from the prior probabilities and likelihoods. A random variable assigns a number to each outcome; its probability distribution leads to the mean (expectation) \(E(X)=\sum x_iP(x_i)\) and variance. The binomial distribution, modelling \(n\) independent Bernoulli trials with success probability \(p\), gives \(P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\).
Conditional probabilityMultiplication theorem and independenceTotal probability and Bayes' TheoremRandom variables and expectationBinomial distribution
Key terms
Conditional probability
\(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\), \(P(B)\neq0\).
Independent events
Events with \(P(A\cap B)=P(A)P(B)\).
Total probability
Summing \(P(E_i)P(A\mid E_i)\) over exhaustive events.
Bayes' Theorem
Reverses conditioning to find \(P(E_i\mid A)\).
Random variable
A function assigning a real number to each outcome.
Binomial distribution
\(P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\).
Extra questions & answers
Try one from this chapter
Conditional probability \(P(A\mid B)\) equals: