)$, we can write:\n\n$$P(Y' \\cap X) = P(X) - P(X \\cap Y)$$\n\nSubstituting our known values:\n\n$$P(Y' \\cap X) = 0.6 - 0.1 = 0.5$$\n\nThis calculation earns you the first mark (**M1**) in the markscheme."},{"type":"text","order":3,"title":"Calculate $P(Y'|X)$","content":"Finally, we substitute our values back into the conditional probability formula we identified in Step 1:\n\n$$P(Y'|X) = \\frac{P(Y' \\cap X)}{P(X)} = \\frac{0.5}{0.6}$$\n\nMultiplying both the numerator and the denominator by 10 to simplify the fraction gives:\n\n$$P(Y'|X) = \\frac{5}{6}$$\n\nThis final substitution and simplification earns the remaining marks (**M1 A1**)."}]},{"id":"venn_diagram","label":"Using a Venn Diagram","steps":[{"type":"text","order":0,"title":"Analyze the question requirements","content":"We need to find the conditional probability $P(Y'|X)$, which is the probability that event $Y$ does not occur, given that event $X$ has occurred.\n\nThe conditional probability formula states:\n\n$$P(Y'|X) = \\frac{P(X \\cap Y')}{P(X)}$$\n\nTo find this, we will construct a Venn diagram representing the probabilities of the events $X$ and $Y$ and use it to visually identify $P(X \\cap Y')$ and $P(X)$."},{"type":"text","order":1,"title":"Find the intersection probability","content":"Before constructing our Venn diagram, we need to know the probability of the intersection, $P(X \\cap Y)$.\n\nUsing the definition of conditional probability:\n\n$$P(X|Y) = \\frac{P(X \\cap Y)}{P(Y)}$$\n\nRearranging this formula allows us to calculate $P(X \\cap Y)$:\n\n$$\\begin{aligned} P(X \\cap Y) &= P(X|Y) \\times P(Y) \\\\ &= 0.4 \\times 0.25 \\\\ &= 0.1 \\end{aligned}$$","highlights":[{"text":"P(Y) = 0.25","location":"specification"},{"text":"P(X|Y) = 0.4","location":"specification"}]},{"type":"text","order":2,"title":"Fill in the Venn diagram","content":"Now we can visualize the probabilities of the regions inside our Venn diagram with two overlapping circles for $X$ and $Y$:\n\n1. **The intersection ($X \\cap Y$):** This is the overlapping middle region. We found this to be $$P(X \\cap Y) = 0.1$$\n2. **The region only in $X$ ($X \\cap Y'$):** Since the entire circle $X$ has a total probability of $0.6$, the region belonging strictly to $X$ (and not $Y$) is:\n\n$$P(X \\cap Y') = P(X) - P(X \\cap Y) = 0.6 - 0.1 = 0.5$$ \n\nThis represents our first mark M1.","highlights":[{"text":"P(X) = 0.6","location":"specification"}]},{"type":"text","order":3,"title":"Apply conditional probability formula","content":"Now we apply the conditional probability formula using the values we've established:\n\n$$P(Y'|X) = \\frac{P(X \\cap Y')}{P(X)}$$\n\nSubstituting the values from our Venn diagram analysis:\n\n$$P(Y'|X) = \\frac{0.5}{0.6}$$\n\nThis calculation earns the second mark M1."},{"type":"text","order":4,"title":"State the final answer","content":"Simplify the fraction to get the final answer:\n\n$$P(Y'|X) = \\frac{5}{6}$$\n\nThis correct simplified fraction earns the final mark A1."}]}],"generatedAt":"2026-06-11T03:10:35.466Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"17bc0ce1-9a7b-4886-86a3-f87f52bda689","label":"c","content":"Find $P(X' \\cup Y)$.","markscheme":"- $P(X' \\cup Y) = 1 - P(X \\cap Y') = 1 - 0.5$ M1\n- $0.5$ A1","marks":2,"order":2,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Method to find the union probability.","code":"M"},{"id":"p2","marks":1,"description":"Final numerical value.","code":"A"}],"finalAnswers":["0.5","1/2"],"generatedAt":"2026-05-06T14:03:41.896Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"complementary_probability","label":"Using Complementary Probability","steps":[{"type":"text","order":0,"title":"Calculate the intersection","content":"First, we can find the probability of the intersection $P(X \\cap Y)$ using the conditional probability formula from the formula booklet:\n\n$$P(X|Y) = \\frac{P(X \\cap Y)}{P(Y)}$$\n\nRearranging this gives:\n\n$$P(X \\ca

Conditional Probability

Question 53

Conditional Probability Questions

Conditional Probability

Question 53

Conditional Probability Questions