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Probability Concepts · Part 3 of 4
9 exam-style questions with model answers, plus 12 quick multi-choice questions — every question on this part of the standard, grouped by the 3 pages of notes they come from.
Write a full answer before you reveal the model one. That comparison is where the learning happens.
In a group of 200 people, 90 own a bike and of those 36 also own a car. Find the probability that a randomly chosen person owns both, and the probability that a bike owner owns a car.
A school of 600 students records that 240 study a language, 150 play in the orchestra, and 84 do both. Construct the two-way table and determine whether studying a language and playing in the orchestra are independent.
A hospital compares two surgeons. Surgeon A succeeded in 210 of 250 operations (84%); Surgeon B succeeded in 150 of 200 (75%). Broken down by case severity: for routine cases Surgeon A succeeded in 190 of 210 (90.5%) and Surgeon B in 90 of 95 (94.7%); for high-risk cases Surgeon A succeeded in 20 of 40 (50%) and Surgeon B in 60 of 105 (57.1%). Discuss what these tables show.
In a group of 100 people, 40 own a cat, 55 own a dog and 15 own both. Draw the Venn diagram regions and find how many own neither.
Of 300 café customers, 168 buy coffee, 96 buy cake and 45 buy both. Find the probability a randomly chosen customer buys exactly one item, and the probability a cake buyer also buys coffee. Interpret the second answer.
A gym reports that of its members, 62% use the weights area, 48% use classes and 35% use the pool. It also reports that 30% use weights and classes, 20% use weights and the pool, 22% use classes and the pool, and 12% use all three. Show that these figures are inconsistent, and explain what may have gone wrong.
A bag holds 4 red and 6 blue counters. Two are drawn without replacement. Find the probability both are red.
On any morning the probability of rain in a coastal town is 0.3. If it rains, the probability the harbour ferry is cancelled is 0.45; if it does not rain, the probability of cancellation is 0.08. Find the probability the ferry runs, and the probability that on a randomly chosen morning it rains and the ferry runs.
A factory has three machines producing the same component: machine X makes 50% of output with a 2% defect rate, machine Y makes 30% with a 4% defect rate, and machine Z makes 20% with a 7% defect rate. A quality manager proposes testing every component from machine Z only, arguing it has the highest defect rate. Evaluate this proposal.