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.

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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]

Two independent fair spinners are used in a game. Spinner $X$ has four equally likely results, $1,2,3,4$ and spinner $Y$ has five equally likely results, $1,2,3,4,5$ The outcome of a round is the ordered pair $(X,Y)$ The table shows the sum for each possible outcome.

$X\backslash Y$	1	2	3	4	5
1	2	3	4	5	6
2	3	4	5	6	7
3	4	5	6	7	8
4	5	6	7	8	9

Let $E$ be the event that the sum is at least 7.

(a)

Find the theoretical probability of $E$

[2]

(b)

Find the probability of the complementary event $E'$

[1]

(c)

Find the expected number of rounds in which $E$ occurs in 80 rounds.

[1]

(d)

In a trial of 80 rounds, $E$ occurs 21 times. Find the relative frequency of $E$

[1]

(e)

Does this trial prove that the spinners are not fair? Give a reason.

[1]

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