## Exam-Style Question on Probability## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 340. This question is similar to one that appeared on an IB Standard paper in 2018. The use of a calculator is allowed.

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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