],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"substitution"}],"connections":[{"to":"substitution","from":"formula","color":"sky","label":"simplify powers"}]},"order":1,"title":"Substitute into the formula","description":"Choose which two throws are successful, then multiply by the probability of two successes and three failures."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$P(X=2)=\\htmlData{anchor=expression}{\\color{@sky}{\\binom{5}{2}(0.3)^2(0.7)^3}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\color{@green}{P(X=2)=0.3087}}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"expression","color":"green","label":"evaluate"}]},"order":2,"title":"Evaluate the probability","description":"Evaluating the binomial expression gives the required probability. This exact value earns the B1 mark."}]}],"generatedAt":"2026-07-20T16:50:47.413Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"a60d3096-b700-4b43-9373-b8132a6d83e6","label":"c","content":"For each person the random variable $Y$ represents the throw on which the first ball lands in the bucket. Using Helen's model, find $P(Y = 5)$ ","markscheme":"$$P(Y=5)$ means 4 misses then 1 hit.\n$0.7^4 \\times 0.3$ M1\n$= 0.07203$ A1","marks":2,"order":2,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correct geometric logic: $(1-p)^4 \\times p$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $0.07203$.","code":"A"}],"finalAnswers":["0.07203"],"generatedAt":"2026-05-12T02:14:06.554Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"ordered_outcomes","label":"Four misses then a hit","steps":[{"type":"text","order":0,"title":"Interpret the event","content":"For $Y=5$, the **first** successful throw must occur on throw $5$. Therefore, throws $1$ to $4$ must all be misses, followed by a hit on throw $5$.","highlights":[{"text":"find $P(Y = 5)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$P(\\text{miss})=1-0.3=0.7$"},{"latex":"$$P(Y=5)=0.7^4\\times0.3$"},{"latex":"$$P(Y=5)=0.07203$"}]},"order":1,"title":"Multiply the probabilities","description":"First find the miss probability, then multiply the probability of four misses by the probability of one hit. Writing the product earns the method mark (M1), and evaluating it correctly earns the accuracy mark (A1)."}]}],"generatedAt":"2026-07-20T16:54:53.773Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"a2ec76b2-49d9-4991-8c40-5d3d429fbd7d","revisionId":"3c77abbc-38ec-46fe-a012-49d8144f5b59","questionNumber":"13","questionType":"LA","specification":"In a game 1, 2, 3, 4 or 5 points can be scored. The random variable $S$ represents the number of points scored.\n\n$$P(S = n) = 0.05 + (n - 1) \\times a$$\n\nwhere $a$ is a constant.","difficulty":2,"difficultyBand":"medium","totalMarks":5,"sourceTitle":"Question Set 6: Statistical Distributions","calculatorUse":null,"parts":[{"id":"5010bd69-92cf-4b3c-9ee7-8d13fdc79a71","label":"0","content":"Find the probability distribution for $S$.","markscheme":"Sum of probabilities $= 1$\n$0.05 + (0.05 + a) + (0.05 + 2a) + (0.05 + 3a) + (0.05 + 4a) = 1$ M1\n$0.25 + 10a = 1 \\implies 10a = 0.75 \\implies a = 0.075$ A1\n\n| $s$ | 1 | 2 | 3 | 4 | 5 |\n|------------|------|-------|-----|-------|------|\n| $P(S = s)$ | 0.05 | 0.125 | 0.2 | 0.275 | 0.35 |\n\n- Correct $a$ A1\n- At least 2 correct probabilities M1\n- Fully correct table A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Sets up equation summing probabilities to 1.","code":"M"},{"id":"p2","marks":2,"description":"Solves for $a = 0.075$.","code":"A"},{"id":"p3","marks":2,"description":"Completes the table with all 5 correct probabilities.","code":"A"}],"finalAn

2.7 Statistical distributions

Question 8

2.7 Statistical distributions Questions

2.7 Statistical distributions

Question 8

2.7 Statistical distributions Questions