The Doctrine of Lapse
First event - Indian Minors
The event can happen to all vassals of the British Empire, which has an Indian primary culture, is uncivilized and has less than 15 prestige. It can additionally only happen to an AI controlled nation and United Kingdom must not have taken the decision The Empress of India.
It has a Mean time to happen of 500 months (roughly 41 years). This will decrease by 10%, if the nation in question has below 10 prestige, another 10% below 5 prestige, above 4 militancy and another 10%, if above 5 militancy. Additionally it will be reduced by 90%, if the government form is not an absolute monarchy.
Second event - British Empire
The British event happens, when the Indian event 'The Doctrine of Lapse' happens. It has two possible options.
Option 1 Excellent! which gives UK 4 prestige and UK will inherit the Indian minor in question.
UK will technically only inherit all the provinces, which has Indian cores and has a militancy of 3 and a consciousness of 1, but since the Indian half of the event gives militancy and consciousness, and vassals seldom annex any non-core territory, it will be a de facto annexation.
Option 2 Bah, more useless territory which gives UK -2 prestige.
From a purely strategic perspective, there is absolutely no reason not to annex the territory. From a tactical perspective, the extra militancy might be undesirable, and in some niche circumstances the drain on Colonial Power may be unwanted.
The Doctrine of Lapse' was a British policy regarding Britain's Indian subject nations, the Princely States, that stated that any such nation could be immediately annexed by the British East India company if the ruler died without a direct heir, or if the ruler was found 'manifestly incompetent'- The doctrine was used historically to annex the states of Satara, Sambalpur, Nagpur, Jhansi and Awadh, the latter being annexation being one of the sparks that ignited the Sepoy rebellion in 1857. The doctrine was repudiated with the Establishment of the British Raj in 1858. Due to the tragic state of affairs that pervades the government of The United Kingdom, the East Indian company today declared that country's existance null and void.