# Do my probabilities make sense?

I am working on a homework problem which I'm not sure if I'm evaluating properly. The problem is as follows:

The article [name here] considered four treatment groups. The groups consisted of patients with different drug therapies:

• Sequential monotherapy (group 1)
• Step-up combination therapy (group 2)
• Initial combination therapy (group 3)
• Initial combination therapy with infliximab (group 4)

Radiographs of hands and feet were used to evaluate disease progression. The number of patients without progression of joint damage was

• 76 of 114 patients (67%)
• 82 of 112 patients (73%)
• 104 of 120 patients (87%)
• 113 of 121 patients (93%)

in groups 1–4, respectively. Suppose that a patient is selected randomly. Let $A$ denote the event that the patient is in group 1, and let $B$ denote the event that there is no progression. Determine the following probabilities:

• a.) $P(A\cup B)$
• b.) $P(A'\cup B')$
• c.) $P(A\cup B')$

It is clear that there are $467$ total patients, and I know that $P(A\cup B) = P(A) + P(B) - P(A\cap B)$

### a.)

My calculated answer is $P = .8843$. My reasoning is that $P(A) = \frac{114}{467}$, $P(B) = \frac{375}{467}$, and $P(A\cap B) = \frac{76}{467}$. Therefore $P(A\cup B) = \frac{114+375-76}{467} = \frac{413}{467}$

### b.)

My calculated answer is $P = .8372$. My reasoning is that $P(A') + P(B') - P(A'\cap B') = \frac{353 + 92 - 54}{467} = \frac{391}{467}$

### c.)

My calculated answer is $P = .3597$. My reasoning is that $P(A) + P(B') - P(A\cap B') = \frac{114 + 92 - 38}{467} = \frac{168}{467}$

I'm sorry if this is not exactly the type of question I should post here but I'm nervous I'm doing these all wrong. Should the probability of the intersection terms above all be in the form of $\frac{N}{467}$? When looking at a smaller subset of numbers the $467$ makes me question my reasoning.

• This all looks fine to me. – JDL Sep 1 '16 at 7:18