estimate the probability that a patient has health insurance, if 59% of Americans have private insurance and 25% have government insurance.

Estimate the probability that it is private insurance

Is this P(A and B) = P(A) x P(B)

P(A and B) represents the probability that both events A and B occurring.

Physically, it means that a person has both private and government insurance, which is unlikely.

The probability of having any insurance is the sum of the two probabilities, i.e.
P(A or B) = P(A) + P(B) - P(A and B)

Assuming anyone having one insurance does not have the other (mutually exclusive), then P(A and B) equals zero, and the result is reduced to
P(A or B) = P(A) + P(B) when A and B are mutually exclusive.