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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Question 1

Paper 2Medium6 marks

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]

Question 2

Paper 1Hard8 marks

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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