Adoctor gives a patient a 6060% chance of surviving bypass surgery after a heart attack. if the patient survives the surgery, then the patient has a 5050% chance that the heart damage will heal. find the probability that the patient survives the surgery and the heart damage heals.

Let BS be the event that the patient survives bypass surgery. Let H be the event that the heart damage will heal. Then P(BS) = 0.60, and also we have a conditional probability: GIVEN that the patient survives, the probability that the heart damage will heal is 0.5, that is P(H|BS) = 0.5 We want to know P(BS and H). Using the formula of the conditional probability: P(H and BS) = P(H|BS)·P(BS) = (0.6)(0.5) = 0.3