Defective batteries: from the normal to the binomial

Problem statement

The supplier of a mobile phone factory provides batteries whose lifetime follows a normal distribution with mean $\mu = 24$ hours and standard deviation $\sigma = 3$ hours. For quality-control purposes, a battery is considered defective if its lifetime is less than $21$ hours.

a) (1 mark) A phone is chosen at random from the production line. Calculate the probability that its battery is considered defective.

b) (1 mark) A distributor receives a batch of $10$ phones. Assuming independence between them, what is the probability that the batch contains at least $9$ phones with a non-defective battery?

Source: Madrid university entrance exam (PAU, Spain) · 2025-2026 · Ordinary session · Mathematics · Question 4.1. Official exam set by the public universities of the Community of Madrid, to whom it belongs; translated from Spanish and reproduced with its source in order to explain it for teaching purposes.