IB Maths AI SL AI 4.3 Probability and distributions questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

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Question 1

Paper 2Easy6 marks

A filling machine produces packets of dried fruit. The mass $X$ in grams, of a randomly selected packet is modelled by a normal distribution with mean $500$ g and standard deviation $4$ g.

(a)

Sketch the normal curve for $X$ Label the horizontal axis with $\mu-\sigma$ $\mu$ $\mu+\sigma$ and their corresponding masses.

[2]

(b)

Use your GDC to calculate the probability that a packet has a mass between $496$ g and $504$ g. Give your answer to four significant figures.

[2]

(c)

Use your GDC to find the mass $x$ such that $90\%$ of packets have a mass less than $x$ Give your answer to the nearest $0.1$ g.

[2]

Question 2

Paper 2Easy5 marks

A game pays the player different amounts on each play. Let $X$ be the player’s net gain, in USD, on one play. The probability distribution of $X$ is shown in Figure 1.

(a)

Write down $P(X=1)$

[1]

(b)

Show that $E(X)=0$

[3]

(c)

Hence state whether the game is fair.

[1]

At a cereal plant, the mass $X$ in grams, of a randomly selected bag is modelled by a normal distribution with mean $502$ g and standard deviation $4.8$ g. A bag is accepted if its mass is at least $500$ g.

The machine can be adjusted by adding the same amount $d$ grams to the mass of every bag. This adjustment does not change the standard deviation.

(a)

Write down the interval that contains approximately $68\%$ of the bag masses at the current setting.

[1]

(b)

Use your GDC to find the probability that a randomly selected bag is accepted at the current setting. Give your answer to four significant figures.

[2]

(c)

The manager requires at least $97.5\%$ of the bags to be accepted. Using your answer to (b), state whether the current setting meets this requirement. Determine the smallest value of $d$ that meets the requirement. Give your answer to three significant figures.

[5]

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