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Probability Distributions · Part 1 of 3
6 exam-style questions with model answers, plus 8 quick multi-choice questions — every question on this part of the standard, grouped by the 2 pages of notes they come from.
Write a full answer before you reveal the model one. That comparison is where the learning happens.
A distribution has P(X=1) = 0.22, P(X=2) = 0.31, P(X=3) = k and P(X=4) = 0.19. Find k, then find P(X ≥ 3).
A café sells a set breakfast in three sizes. Records show 45% of customers choose small ($12), 35% medium ($16) and 20% large ($21). Let X be the amount a randomly chosen customer spends. Write the distribution, find P(X > 12), and explain whether X is discrete or continuous.
A charity raffle sells 500 tickets at $5 each. There is one prize of $600, three prizes of $100 and ten prizes of $20. A buyer says: 'With 14 prizes among 500 tickets, my chance of winning something is nearly 3%, and the prizes total $1 100, so the raffle is roughly fair.' Discuss the buyer's reasoning.
A random variable has P(X=0) = 0.3, P(X=1) = 0.5, P(X=2) = 0.2. Find E(X) and the standard deviation.
A tour operator charges $85 per passenger and has fixed costs of $900 per trip. Passenger numbers per trip have mean 18 and standard deviation 4. Find the mean and standard deviation of profit, and comment on the risk.
A courier company's parcels have mass with mean 2.4 kg and standard deviation 0.9 kg. A van carries 60 randomly selected parcels. A manager calculates the total load's standard deviation as 60 × 0.9 = 54 kg and concludes the load is far too unpredictable to plan around. Discuss.