-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,"isCalculation":true,"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"}],"finalAnswers":[],"generatedAt":"2026-05-12T02:14:06.554Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"normalise_then_substitute","label":"Normalise then substitute","steps":[{"type":"text","order":0,"title":"Use total probability","content":"A probability distribution lists every possible value of $S$ and its probability. Because the possible scores are $1,2,3,4,5$, their probabilities must add to $1$.","highlights":[{"text":"In a game 1, 2, 3, 4 or 5 points can be scored.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=prob_sum}{\\color{@sky}{P(S=1)+P(S=2)+P(S=3)+P(S=4)+P(S=5)}}=1$"},{"latex":"$$\\htmlData{anchor=expanded_sum}{\\color{@sky}{0.05+(0.05+a)+(0.05+2a)+(0.05+3a)+(0.05+4a)}}=1$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"prob_sum"},{"color":"sky","style":"text-color","anchor":"expanded_sum"}],"connections":[{"to":"expanded_sum","from":"prob_sum","color":"sky","label":"substitute each score","style":"solid"}]},"order":1,"title":"Form the equation","description":"Substitute each possible score into the given formula, then use the fact that the probabilities sum to 1. Writing this equation earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=amber_expression}{\\color{@amber}{0.05+(0.05+a)+(0.05+2a)+(0.05+3a)+(0.05+4a)}}=1$"},{"latex":"$$\\htmlData{anchor=amber_simplified}{\\color{@amber}{0.25+10a}}=1$"},{"latex":"$$10a=\\htmlData{anchor=sky_difference}{\\color{@sky}{0.75}}$"},{"latex":"$$a=\\htmlData{anchor=green_value}{\\color{@green}{0.075}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"amber_expression"},{"color":"amber","style":"text-color","anchor":"amber_simplified"},{"color":"sky","style":"text-color","anchor":"sky_difference"},{"color":"green","style":"text-color","anchor":"green_value"}],"connections":[{"to":"amber_simplified","from":"amber_expression","color":"amber","label":"collect like terms","style":"dashed"},{"to":"sky_difference","from":"amber_simplified","color":"sky","label":"subtract 0.25","style":"dashed"},{"to":"green_value","from":"sky_difference","color":"green","label":"divide by 10","style":"dashed"}]},"order":2,"title":"Solve for the constant","description":"Collect like terms and solve the normalisation equation. Obtaining 0.25 + 10a = 1 earns the algebraic A1, and obtaining a = 0.075 earns the separate correct-value A1. The later probability work earns M1 for at least two correct probabilities, and the fully correct distribution earns the final A1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$P(S=n)=0.05+(n-1)(0.075)$"},{"l

Statistical Distributions

Question 8

Statistical Distributions Questions

Statistical Distributions

Question 8

Statistical Distributions Questions