What is the role of Bayes’ Theorem in Class 12 Probability?

<p>Bayes' Theorem results in a formula for conditional Probability which is used to determine the conditional probability of a mutually inclusive event. We have includes usage of this theorem in our Probability solutions in right places. Students can dig deeper into this theorem and learn using its formula through the NCERT Solutions provided by shiksha.</p>

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<p>Bayes' Theorem results in a formula for conditional Probability which is used to determine the&nbsp;conditional probability of a mutually inclusive event. As per NCERT definition,"<strong>If </strong><math ><semantics><mrow><msub><mi>B</mi><mn>1</mn></msub><mo separator="true">, </mo><msub><mi>B</mi><mn>2</mn></msub><mo separator="true">, </mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">, </mo><msub><mi>B</mi><mi>n</mi></msub></mrow>&lt;annotation encoding="application/x-tex"&gt;B_1, B_2, ., B_n</annotation></semantics></math><strong><span aria-hidden="true">? </span> are mutually exclusive and exhaustive events and </strong><math ><semantics><mrow><mi>A</mi></mrow>&lt;annotation encoding="application/x-tex"&gt;A</annotation></semantics></math><strong> is any event that occurred, then the probability of </strong><math ><semantics><mrow><msub><mi>B</mi><mi>i</mi></msub></mrow>&lt;annotation encoding="application/x-tex"&gt;B_i</annotation></semantics></math><strong><span aria-hidden="true">Bi? </span> given that </strong><math ><semantics><mrow><mi>A</mi></mrow>&lt;annotation encoding="application/x-tex"&gt;A</annotation></semantics></math><strong><span aria-hidden="true">A</span> has occurred is given by:</strong></p><p><math ><semantics><mrow><mi>P</mi><mo stretchy="false"> (</mo><msub><mi>B</mi><mi>i</mi></msub><mi mathvariant="normal">? </mi><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false"> (</mo><msub><mi>B</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>? </mo><mi>P</mi><mo stretchy="false"> (</mo><mi>A</mi><mi mathvariant="normal">? </mi><msub><mi>B</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><munderover><mo>? </mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false"> (</mo><msub><mi>B</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>? </mo><mi>P</mi><mo stretchy="false"> (</mo><mi>A</mi><mi mathvariant="normal">? </mi><msub><mi>B</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow>&lt;annotation encoding="application/x-tex"&gt;P (B_i|A) = \frac {P (B_i) \cdot P (A|B_i)} {\sum_ {k=1}^ {n} P (B_k) \cdot P (A|B_k)}</annotation></semantics></math></p><p data-start="875" data-end="942">As per NCERT Bayes' Theorem is used in problems like source of a defective product from different venues, and mainly classification problems.</p>

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