## Exam-Style Questions.## Problems adapted from questions set for previous Mathematics exams. |

## 1. | IB Standard |

A random variable \(R\) has the probability distribution as shown in the following table:

\(r\) | 1 | 2 | 3 | 4 | 5 |

\(P(R=r)\) | 0.2 | a | b | 0.25 | 0.15 |

(a) Given that \(E(R) = 2.85\) find \(a\) and \(b\).

(b) Find \( P(R \gt 2 )\).

## 2. | IB Analysis and Approaches |

Amelia selects a disc from a bag containing 5 green discs, 4 blue discs and 3 red discs.

She wins £1 for a green disc, £3 for a blue disc, and £7 for a red disk. The game costs £4 to play.

(a) Calculate Amelia's expected gain for this game, and hence show that the game is not fair.

After a while, \(n\) red discs are added and Amelia wins £5 if she selects one. The game is now fair.

(b) Create an equation to represent this information.

(c) Hence, calculate the value of \(n\).

## 3. | IB Applications and Interpretation |

A game is played with a biased five-sided spinner. The possible scores, \(X\) and their probabilities are shown in the following table.

Score \(x\) | -5 | -1 | 0 | 2 | 10 |

\(P(X=x)\) | 0.2 | p | 0.3 | 0.3 | 0.1 |

(a) Find the exact value of p.

Jaedee plays the game once.

(b) Calculate the expected score.

Jaedee plays the game twice and adds the two scores together.

(c) Find the probability Jaedee has a total score of 10.

## 4. | IB Analysis and Approaches |

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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