# Exam-Style Question on Probability

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 562. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is not allowed.

A red spinner is designed with five possible outcomes. Let $$X$$ be the score obtained when the spinner is spun. The probability distribution for $$X$$ is given in the following table.

 $$x$$ 3 4 5 8 10 $$P(X=x)$$ $$p$$ $$2p$$ $$p$$ $$2p$$ $$p$$

(a) Find the value of p.

(b) Find the value of $$E(x)$$.

A blue spinner is also designed with five possible outcomes. Let $$Y$$ be the score obtained when the spinner is spun. The probability distribution for $$Y$$ is given in the following table.

 $$y$$ 1 2 3 4 5 $$P(Y=y)$$ $$r$$ $$r$$ $$q$$ $$q$$ $$q$$

(c) State the range of possible values of $$q$$.

(d) State the range of possible values of $$r$$.

(e) Hence find the range of possible values of $$E(Y)$$.

Arnold and Bernie play a game using these spinners. Arnold spins the red spinner once and Bernie spins the blue spinner once. The probability that Arnold's score is less than Bernie's score is $$\frac{1}{7}$$.

(f) Find the value of $$E(Y)$$.

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