That would be an example of argument from anecdote, where a particular instance is taken to be representative of the whole. This is usually an event experienced personally, or by a friend or family member, or a story which receives heavy media attention, and thus has a strong influence over the reasoning of the person in question.

We can see the utility of vaccines when we look at the aggregate statistics, and people who do not receive them will be at greater danger, but this does not mean that every unvaccinated individual will become unhealthy.
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