The probability that no one in the family of 4 will be hospitalized during a year is 0.52
Step-by-step explanation:
The probability for a person to be hospitalized during one year is

As a consequence, the probability for a person of NOT being hospitalized during one year is

If we have 4 people in the family, the probability of each of them of NOT being hospitalized during the year is independent from the probability of the others - therefore, these are independent events. This means that the total probability of the 4 people of not being hospitalized during a year is:

So, the probability that no one in the family of 4 will be hospitalized during a year is 0.52.
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