https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-dependent/v/independent-events-1 2021-05-19T21:45:24Z Independent & dependent probability This time around we're not going to tell you whether we're working on a dependent or independent probability event problem. You tell us! Sal Khan video https://cdn.kastatic.org/googleusercontent/H4xO9_buopunL7VGOca9o2oSfBtxSSk_xpeSU-D0u-GONjPQCy7GY36xzDXPiIvdvMM69NBCyOVKgTHttafcaEs Independent & dependent probability This time around we're not going to tell you whether we're working on a dependent or independent probability event problem. You tell us! https://cdn.kastatic.org/ka-youtube-converted/Za7G_eWKiF4.mp4/Za7G_eWKiF4.mp4 159 Independent events https://www.khanacademy.org/math/mr-class-10/x5cfe2ca097f0f62c:probability/x5cfe2ca097f0f62c:finding-probability-of-compound-events/v/independent-events-example-test-taking-indian-accent 2025-06-03T19:25:43.416794683Z Independent events example: test taking Have you ever taken a test and discovered you have no choice but to guess on a couple of problems? In this example problem, we are considering the probability of two independent events occurring. Skyloom (Dubbing) video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Independent events example: test taking Have you ever taken a test and discovered you have no choice but to guess on a couple of problems? In this example problem, we are considering the probability of two independent events occurring. https://cdn.kastatic.org/ka-youtube-converted/DtQzDcIQYsw.mp4/DtQzDcIQYsw.mp4 236 Independent events https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:prob-comb/x9e81a4f98389efdf:compound-probability-of-ind-events-using-mult-rule/v/interpreting-general-multiplication-rule 2022-10-07T00:22:18.752150461Z Interpreting general multiplication rule We can express the probability that two events both occur symbolically using the general multiplication rule, and we can interpret probability statements that are expressed symbolically. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/12debcfa-c173-42b7-bffd-810bcfde0794_1280_720_base.png Interpreting general multiplication rule We can express the probability that two events both occur symbolically using the general multiplication rule, and we can interpret probability statements that are expressed symbolically. https://cdn.kastatic.org/ka-youtube-converted/yTJawPRzCzI.mp4/yTJawPRzCzI.mp4 291 Independent events https://www.khanacademy.org/math/statistics-probability/probability-library/multiplication-rule-independent/e/independent_probability 2025-04-08T07:26:09.960930891Z Independent probability Find probabilities of independent events like flipping a heads and rolling an even number! Ben Eater exercise https://www.khanacademy.org/math/class-12-tg/x6469e28db15923ec:probability/x6469e28db15923ec:independent-and-dependent-events-conditional-probability-multiplication-theorem-and-baye-s-theorem-part-2/v/testing-for-independence 2025-06-01T14:50:00.385158177Z Testing for independence This video focuses on how to test whether two events are independent. We'll use the core condition for independence: events E and F are independent if and only if the probability of E and F both occurring, P(E intersect F), is equal to the product of their individual probabilities, P(E) times P(F). We'll apply this test to various scenarios, including given abstract probabilities and outcomes from throwing dice. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Testing for independence This video focuses on how to test whether two events are independent. We'll use the core condition for independence: events E and F are independent if and only if the probability of E and F both occurring, P(E intersect F), is equal to the product of their individual probabilities, P(E) times P(F). We'll apply this test to various scenarios, including given abstract probabilities and outcomes from throwing dice. https://cdn.kastatic.org/ka-youtube-converted/Xm8421Vtfyo.mp4/Xm8421Vtfyo.mp4 437 Independent events https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:independent-events/e/testing-for-independence 2025-08-08T02:45:51.78671331Z Testing for independence Testing for independence Ashish Gupta exercise https://www.khanacademy.org/math/class-12-tg/x6469e28db15923ec:probability/x6469e28db15923ec:independent-and-dependent-events-conditional-probability-multiplication-theorem-and-baye-s-theorem-part-2/v/finding-probabilities-given-independence 2025-06-01T14:50:00.385158177Z Finding probabilities given independence In this video, we explore how to calculate various probabilities when two events, A and B, are given to be independent. Using the property P(A intersect B) = P(A)P(B), we will find probabilities of events such as P(A and B), P(A or B), the conditional probabilities P(A|B) and P(B|A), P(A and not B), and P(neither A nor B). Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Finding probabilities given independence In this video, we explore how to calculate various probabilities when two events, A and B, are given to be independent. Using the property P(A intersect B) = P(A)P(B), we will find probabilities of events such as P(A and B), P(A or B), the conditional probabilities P(A|B) and P(B|A), P(A and not B), and P(neither A nor B). https://cdn.kastatic.org/ka-youtube-converted/eL-f6jXZIRM.mp4/eL-f6jXZIRM.mp4 308 Independent events https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:independent-events/e/finding-probabilities-given-independence 2025-08-08T02:45:51.78671331Z Finding probabilities given independence Finding probabilities given independence Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:independent-events/v/proving-statements-for-independent-events 2025-06-01T14:50:00.385158177Z Proving statements for independent events This video focuses on proving important theoretical statements related to independent events. First, we will prove that if two events E and F are independent, then event E and the complement of F (F prime) are also independent. Second, we'll prove that if A and B are independent events, the probability of at least one of them occurring is given by 1 minus (the probability of A not occurring times the probability of B not occurring). Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Proving statements for independent events This video focuses on proving important theoretical statements related to independent events. First, we will prove that if two events E and F are independent, then event E and the complement of F (F prime) are also independent. Second, we'll prove that if A and B are independent events, the probability of at least one of them occurring is given by 1 minus (the probability of A not occurring times the probability of B not occurring). https://cdn.kastatic.org/ka-youtube-converted/F6AERK_kSqk.mp4/F6AERK_kSqk.mp4 253 Independent events https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:independent-events/v/independent-events-dice-thrown-and-balls-picked 2025-06-01T14:50:00.385158177Z Independent events (dice thrown and balls picked) In this video, we apply the concept of independent events to solve practical probability problems. We'll first tackle a problem involving tossing a die three times and finding the probability of getting an odd number at least once, by considering the complementary event. Then, we'll analyze a scenario where two balls are drawn at random with replacement from a box containing black and red balls, and calculate probabilities for different outcomes like both balls being red, or one being black and the other red. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Independent events (dice thrown and balls picked) In this video, we apply the concept of independent events to solve practical probability problems. We'll first tackle a problem involving tossing a die three times and finding the probability of getting an odd number at least once, by considering the complementary event. Then, we'll analyze a scenario where two balls are drawn at random with replacement from a box containing black and red balls, and calculate probabilities for different outcomes like both balls being red, or one being black and the other red. https://cdn.kastatic.org/ka-youtube-converted/Y5jIO9f_8zU.mp4/Y5jIO9f_8zU.mp4 276 Independent events https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:independent-events/v/independent-events-other-scenarios 2025-06-01T14:50:00.385158177Z Independent events (other scenarios) This video covers more application scenarios involving independent and conditional probabilities. We'll analyze a problem where two individuals, A and B, try to solve a problem independently, and calculate the probability that the problem is solved and the probability that exactly one of them solves it. Another example involves students' newspaper reading habits, where we find probabilities such as a student reading neither Hindi nor English newspapers, or reading English given they read Hindi. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Independent events (other scenarios) This video covers more application scenarios involving independent and conditional probabilities. We'll analyze a problem where two individuals, A and B, try to solve a problem independently, and calculate the probability that the problem is solved and the probability that exactly one of them solves it. Another example involves students' newspaper reading habits, where we find probabilities such as a student reading neither Hindi nor English newspapers, or reading English given they read Hindi. https://cdn.kastatic.org/ka-youtube-converted/SRUbP25ser8.mp4/SRUbP25ser8.mp4 373 Independent events https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:probability-ncert-new/x6f66cbee9a2f805b:independent-events/e/finding-probabilities-of-independent-events 2025-08-08T02:45:51.78671331Z Finding probabilities of independent events Finding probabilities of independent events Ashish Gupta exercise