Question

1.14. Problem. (Section 3.4) A public health researcher examines the medical records of a group of 937 men who died in 1999 and discovers that 210 of the men died from causes related to heart disease. Moreover, 312 of the 937 men had at least one parent who suffered from heart disease, and, of these 312 men, 102 died from causes related to heart disease. Determine the probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease. Upon arrival at a hospital’s emergency room, patients are categorized according to their condition as critical, serious, or stable. In the past year,

• 10% of ER patients were critical

• 30% of ER patients were serious

• 60% of ER patients were stable

• 40% of critical patients died.

• 10% of the serious patient died.

• 1% of the stable patients died. Given a patient survived, what is the probability that they were categorized as serious upon arrival?

Answer 1

1.14

Let H be the event that the men died from causes related to heart disease.

Let P be the event that men had at least one parent who suffered from heart disease. And ~P be the event that neither of parent who suffered from heart disease

Total Number of Men, n(U) = 937

Number of men died from causes related to heart disease = n(H) = 210

Number of men had at least one parent who suffered from heart disease = n(P) = 312

Number of men for whom neither of parent suffered from heart disease = n(~p) = n(U) - n(P) = 937 - 312 = 625

Number of men died from causes related to heart disease given that at least one parent who suffered from heart disease

= n(H and P) = 102

Number of men died from causes related to heart disease and neither of parent who suffered from heart disease

= n(H and ~P) = n(H) - n(H and P) = 210 - 102 = 108

Probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease. = P(H | ~P) = n(H and ~P) / n(~P)

= 108 / 625

= 0.1728

Let C, S and ST be the event that the ER patients are critical, serious and stable respectively.

Let D be the event that the patients died. Then ~D be the event that the patient survived.

P(C) = 0.1

P(S) = 0.3

P(ST) = 0.6

P(D | C) = 0.4

P(D | S) = 0.1

P(D | ST) = 0.01

By law of total probability,

P(D) = P(C) P(D | C) + P(S) P(D | S) + P(ST) P(D | ST)

= 0.1 * 0.4 + 0.3 * 0.1 + 0.6 * 0.01

= 0.076

Probability that the patient is survived, P(~D) = 1 - P(D) = 1 - 0.076 = 0.924

Also, P(~D | S) = 1 - P(D | S) = 1 - 0.1 = 0.9

Given a patient survived, what is the probability that they were categorized as serious upon arrival = P(S | ~D)

= P(~D | S) * P(S) / P(~D)

= 0.9 * 0.3 / 0.924

= 0.2922078