649e7a284a77","label":"b","content":"Work out the probability that Hannah will win both games.","markscheme":"- $0.6 \\times 0.7$ M1\n- $0.42$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Multiplies the probability of winning chess by the probability of winning backgammon.","code":"M"},{"id":"p2","marks":1,"description":"Final answer 0.42.","code":"A"}],"finalAnswers":["0.42"],"generatedAt":"2026-05-06T11:07:17.419Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"multiply_probabilities","label":"Multiply the probabilities","steps":[{"type":"text","order":0,"title":"Identify the events","content":"“Both games” means Hannah must win the chess game **and** win the backgammon game. For independent events, meaning one result does not affect the other, multiply their probabilities.","highlights":[{"text":"win both games","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$P(\\text{both wins})=\\htmlData{anchor=product}{\\color{@sky}{0.6\\times0.7}}$"},{"latex":"$$P(\\text{both wins})=\\htmlData{anchor=result}{\\color{@green}{0.42}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"product"},{"color":"green","style":"text-color","anchor":"result"}],"connections":[{"to":"result","from":"product","color":"sky","label":"multiply"}]},"order":1,"title":"Multiply the probabilities","description":"Use the chess probability and the backgammon probability. Writing 0.6 × 0.7 earns M1, and obtaining 0.42 earns A1."}]}],"generatedAt":"2026-07-20T10:44:38.074Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"0b3cbef6-7491-4771-bc32-3866d848d2a6","revisionId":"d75113e0-58be-4bac-89a6-1f2c096bad75","questionNumber":null,"questionType":"LA","specification":"A jar contains 8 silver coins and 2 gold coins.\n\nA second jar contains 5 silver coins and 6 gold coins.\n\nLila picks one coin at random from the first jar.\n\nShe then picks one coin at random from the second jar.","difficulty":2,"difficultyBand":"hard","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"4f6c6210-059e-48ac-9dfe-2ada7099708d","label":"a","content":"Work out the probability that Lila picks two silver coins.","markscheme":"- $P(\\text{Silver}_1) = \\frac{8}{10}$ and $P(\\text{Silver}_2) = \\frac{5}{11}$ B1\n- $\\frac{8}{10} \\times \\frac{5}{11}$ M1\n- $\\frac{40}{110}$ or $\\frac{4}{11}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly identifying the probabilities of selecting a silver coin from each jar (8/10 and 5/11).","code":"B"},{"id":"p2","marks":1,"description":"Multiplying the probabilities of the independent events.","code":"M"},{"id":"p3","marks":1,"description":"Calculating the final probability correctly.","code":"A"}],"finalAnswers":["4/11","40/110","0.3636..."],"generatedAt":"2026-05-06T14:06:11.852Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"multiply_independent_probabilities","label":"Multiply the probabilities","steps":[{"type":"text","order":0,"title":"Find each probability","content":"The first jar has $8+2=10$ coins, so $P(\\text{Silver}_1)=\\frac{8}{10}$. The second jar has $5+6=11$ coins, so $P(\\text{Silver}_2)=\\frac{5}{11}$. Stating both correct probabilities earns **B1**.","highlights":[{"text":"A jar contains 8 silver coins and 2 gold coins.","location":"specification"},{"text":"A second jar contains 5 silver coins and 6 gold coins.","location":"specification"}]},{"type":"text","order":1,"title":"Multiply for both","content":"Lila must get silver from the first jar **and** silver from the second. For two successive independent picks, multiply their probabilities:\n\n$$\\frac{8}{10}\\times\\frac{5}{11}$$\n\nWriting this multiplication earns **M1**.","highlights":[{"text":"Work out the probability that Lila picks two silver coins.","lo

Probability Trees

Question 13

Probability Trees Questions

Probability Trees

Question 13

Probability Trees Questions