le":"2.3.1 Mutually exclusive and independent events Flashcards","topicId":"2f8589ed-0c43-42e0-b3b6-b2530f42b644","topicSlug":"mutually-exclusive-and-independent-events","cards":[{"id":"b7f122af-bc7c-4596-94ab-9163fca61d0a","cardType":"cloze","front":"Events $A$ and $B$ are mutually exclusive when $P(A\\cap B)=$ $\\text{[...]}$.","back":"Events $A$ and $B$ are mutually exclusive when $P(A\\cap B)=$ $\\boxed{0}$.","points":null,"mcqOptions":null},{"id":"a502e9fb-5c82-4803-901b-0f01b0969021","cardType":"basic","front":"Which operation is suggested by “$A$ or $B$” in probability?","back":"Use the union $A\\cup B$.","points":1,"mcqOptions":null},{"id":"242669e2-76e0-4306-b356-07a1aa8ce996","cardType":"basic","front":"Which operation is suggested by “$A$ and $B$” in probability?","back":"Use the intersection $A\\cap B$.","points":1,"mcqOptions":null},{"id":"c59da2c8-005d-4a5c-9685-1a67ace25e53","cardType":"cloze","front":"The general addition rule is $\\text{[...]}$.","back":"The general addition rule is $\\boxed{P(A\\cup B)=P(A)+P(B)-P(A\\cap B)}$.","points":null,"mcqOptions":null},{"id":"aaeaad4f-3a29-47d8-b76a-acd535185f6c","cardType":"basic","front":"Why is $P(A\\cap B)$ subtracted in the general addition rule?","back":"The intersection was counted twice in $P(A)+P(B)$.","points":2,"mcqOptions":null},{"id":"0d635e47-13d0-4999-9521-f9519ac4aebc","cardType":"basic","front":"How does the addition rule simplify when $A$ and $B$ are mutually exclusive?","back":"$P(A\\cup B)=P(A)+P(B)$","points":1,"mcqOptions":null},{"id":"687c65b5-29c7-4df2-bf6d-59f0b160a838","cardType":"basic","front":"A die event $A=\\{2\\}$ and event $B=\\{5,6\\}$. Find $P(A\\cup B)$.","back":"$\\frac12$","points":1,"mcqOptions":null},{"id":"0ccb84ef-b247-47ad-b1f8-91582a3321cf","cardType":"basic","front":"Which addition rule must be used if two events may overlap?","back":"Use the general addition rule, subtracting $P(A\\cap B)$.","points":1,"mcqOptions":null},{"id":"354c3793-2b41-4858-a559-e295f6c31e16","cardType":"cloze","front":"Independent events satisfy $\\text{[...]}$.","back":"Independent events satisfy $\\boxed{P(A\\cap B)=P(A)P(B)}$.","points":null,"mcqOptions":null},{"id":"6452897d-4ccc-4edc-b625-2867b2de27ba","cardType":"basic","front":"If $P(B)>0$, which conditional-probability equation tests whether $A$ and $B$ are independent?","back":"$P(A\\mid B)=P(A)$","points":1,"mcqOptions":null},{"id":"6a081b07-400d-44e6-9b43-75a0a502e7e3","cardType":"basic","front":"How is independence shown on a probability tree?","back":"Probabilities remain unchanged at the next stage.","points":1,"mcqOptions":null},{"id":"fa21bdcb-4d73-485a-9682-2c14240ac662","cardType":"basic","front":"Given $P(A)=0.4$, $P(B)=0.7$ and $P(A\\cap B)=0.28$, are the events independent?","back":"Yes, since $0.4\\times0.7=0.28$.","points":2,"mcqOptions":null},{"id":"841c641b-873c-4d2b-af18-9768928a4518","cardType":"basic","front":"A biased coin with $P(H)=0.6$ is tossed twice independently. Find $P(HT)$.","back":"$0.24$","points":1,"mcqOptions":null},{"id":"1aba7744-76df-4b40-ad9a-12220bce7dc5","cardType":"basic","front":"A biased coin with $P(H)=0.6$ is tossed twice independently. Find the probability of exactly one head.","back":"$0.48$, from $P(HT)+P(TH)=0.24+0.24$.","points":2,"mcqOptions":null},{"id":"5897dac0-9598-406d-a400-fd6b1c4f8d6e","cardType":"basic","front":"Why can two mutually exclusive events with positive probabilities not be independent?","back":"Mutual exclusivity gives $P(A\\cap B)=0$, but independence requires $P(A)P(B)>0$.","points":2,"mcqOptions":null},{"id":"d747bb2e-3e8c-43ab-9ab5-88348aceff63","cardType":"basic","front":"When can two events be both mutually exclusive and independent?","back":"When at least one event has probability zero.","points":1,"mcqOptions":null},{"id":"2307d03f-e48d-4f1d-9ae7-777d830b5a19","cardType":"cloze","fron