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

## 1. | IB Standard |

The weights of players in a sports league are normally distributed with a mean of 75.2 kg, (correct to three significant figures). It is known that 75% of the players have weights between 67 kg and 80 kg. The probability that a player weighs less than 67 kg is 0.05.

(a) Find the probability that a player weighs more than 80 kg.

(b) Write down the standardized value, z, for 67 kg.

(c) Hence, find the standard deviation of weights.

To take part in a tournament, a player's weight must be within 1.5 standard deviations of the mean.

(d) Find the set of all possible weights of players that take part in the tournament.

(e) A player is selected at random. Find the probability that the player takes part in the tournament.

Of the players in the league, 22% are women. Of the women, 60% take part in the tournament.

(f) Given that a player selected at random takes part in the tournament, find the probability that the selected player is a woman.

## 2. | IB Analysis and Approaches |

The length, \(X\) minutes , of a certain category of online video is normally distributed with a mean of 28.

The probability that \(X\) is less than 20 is 0.213.

(a) Find \(P(20 \lt X \lt 28)\).

(b) Find the standard deviation of \(X\).

(c) Hence, find the probability that a video selected at random from this category lasts longer than 33 minutes

A random sample of 12 videos from this category are downloaded.

(d) How many of these videos could be expected to last longer than 33 minutes?

(e) Find the probability that exactly two of these videos last longer than 33 minutes.

(f) A video selected at random from the complete online collection of videos in this category has a running time of less than 20 minutes. Find the probability that its length is between ten and fifteen minutes.

## 3. | IB Standard |

The length of Costlow's bâtard bread loaves in centimetres is normally distributed with mean \( \mu \). The following table shows probabilities for values of \(L\).

Length (\(L\)) | \(L \lt 30\) | \(30 \le L \le 42\) | \(L \gt 42\) |
---|---|---|---|

P(X) | 0.07 | 0.86 | \(k\) |

(a) Calculate the value of \(k\).

(b) Show that \( \mu \) = 36.

(c) Find P(\(L \gt 39\)).

The loaves are displayed in baskets of twelve. Any loaves with a length less than 31cm are classified as short.

(d) Find the probability that a basket of loaves selected at random contains at most one loaf that is short.

(e) Each Costlow supermarket has 40 baskets of loaves. One of the Costlow supermarkets is selected at random. Find the expected number of baskets in this supermarket that contain at most one loaf that is too short.

(f) Find the probability that at least 28 baskets in this supermarket contain at most one loaf that is too short.

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