https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:bayes-theorem-proofs/v/proving-bayes-theorem 2025-06-01T14:50:00.385158177Z Proving Bayes theorem In this video, we provide a formal, step-by-step proof of Bayes' theorem. We begin by explaining the concept of a partition of a sample space. Then, we derive the crucial Theorem of Total Probability, which expresses the probability of an event A in terms of its conditional probabilities on these partitions. Finally, by substituting the Theorem of Total Probability into the basic definition of conditional probability, we arrive at the complete formula for Bayes' theorem. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Proving Bayes theorem In this video, we provide a formal, step-by-step proof of Bayes' theorem. We begin by explaining the concept of a partition of a sample space. Then, we derive the crucial Theorem of Total Probability, which expresses the probability of an event A in terms of its conditional probabilities on these partitions. Finally, by substituting the Theorem of Total Probability into the basic definition of conditional probability, we arrive at the complete formula for Bayes' theorem. https://cdn.kastatic.org/ka-youtube-converted/UKBWHnyXSwM.mp4/UKBWHnyXSwM.mp4 340 Bayes' theorem (proofs) https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:bayes-theorem-proofs/v/bayes-theorem-tree-diagrams-and-notations 2025-06-01T14:50:00.385158177Z Bayes theorem - tree diagrams and notations This video bridges the gap between the intuitive tree diagram method and the formal notation of Bayes' theorem. We'll take a classic "balls in bags" problem and solve it first using a simple tree diagram to visualize the probabilities. Then, we will define the events using formal notation (E1, E2, A) and show how each number from our tree diagram corresponds directly to a term in the Bayes' theorem formula, P(E2|A) = P(E2)P(A|E2) / [P(E1)P(A|E1) + P(E2)P(A|E2)]. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Bayes theorem - tree diagrams and notations This video bridges the gap between the intuitive tree diagram method and the formal notation of Bayes' theorem. We'll take a classic "balls in bags" problem and solve it first using a simple tree diagram to visualize the probabilities. Then, we will define the events using formal notation (E1, E2, A) and show how each number from our tree diagram corresponds directly to a term in the Bayes' theorem formula, P(E2|A) = P(E2)P(A|E2) / [P(E1)P(A|E1) + P(E2)P(A|E2)]. https://cdn.kastatic.org/ka-youtube-converted/s86CfOGB8ic.mp4/s86CfOGB8ic.mp4 287 Bayes' theorem (proofs) https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:bayes-theorem-proofs/v/bayes-theorem-proving-independence 2025-06-01T14:50:00.385158177Z Bayes theorem - proving independence In this video, we work through an interesting proof related to conditional probability. Consider an urn with 'm' white and 'n' black balls. A ball is drawn, its color is noted, and it's returned to the urn along with 'k' additional balls of the same color. We will then calculate the total probability of drawing a white ball on the second draw and show, perhaps surprisingly, that this probability is independent of 'k', the number of additional balls added. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Bayes theorem - proving independence In this video, we work through an interesting proof related to conditional probability. Consider an urn with 'm' white and 'n' black balls. A ball is drawn, its color is noted, and it's returned to the urn along with 'k' additional balls of the same color. We will then calculate the total probability of drawing a white ball on the second draw and show, perhaps surprisingly, that this probability is independent of 'k', the number of additional balls added. https://cdn.kastatic.org/ka-youtube-converted/s6mT-023vOQ.mp4/s6mT-023vOQ.mp4 318 Bayes' theorem (proofs) https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:bayes-theorem-proofs/v/bayes-theorem-solving-an-equation 2025-06-01T14:50:00.385158177Z Bayes theorem - solving an equation In this video, we tackle a unique problem where we use a given probability to solve for an unknown variable in the initial setup. A bag contains a mix of two-headed coins and fair coins, with the number of each defined by a variable 'n'. Given that the total probability of picking a coin and tossing a head is 31/42, we set up an equation using the law of total probability and solve it to determine the value of 'n'. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Bayes theorem - solving an equation In this video, we tackle a unique problem where we use a given probability to solve for an unknown variable in the initial setup. A bag contains a mix of two-headed coins and fair coins, with the number of each defined by a variable 'n'. Given that the total probability of picking a coin and tossing a head is 31/42, we set up an equation using the law of total probability and solve it to determine the value of 'n'. https://cdn.kastatic.org/ka-youtube-converted/SE0BfRU58Og.mp4/SE0BfRU58Og.mp4 268 Bayes' theorem (proofs) https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:bayes-theorem-proofs/e/baye-s-theorem-solving-an-equation 2025-08-08T02:45:51.78671331Z Baye's theorem - solving an equation Baye's theorem - solving an equation Ashish Gupta exercise