Over the following three months, 300 convicted low-level drug offenders were assigned to different levels of treatment utilizing the methodology described above.
$$\begin{array} { | l | l | l | l | l | }
\hline & \text { Extensive treatment } & \text { Conventional treatment } & \text { No treatment } & \text { Total } \
\hline \text { Female } & 41 & 33 & 51 & 125 \
\hline \text { Male } & 59 & 67 & 49 & 175 \
\hline \text { Total } & 100 & 100 & 100 & 300 \
\hline
\end{array}$$ As the table indicates, we have a total of 300 inmates who were assigned to extensive treatment, conventional treatment, or no treatment according to the methodology presented in question 1. Of the 125 females, 51 have been assigned to no treatment, 33 to conventional treatment, and 41 to extensive treatment. The remaining 175 offenders were male, 49 who were assigned to no treatment, 67 to conventional treatment, and 59 to extensive treatment.
a. If we selected one individual at random from the sample, what is the probability that a person is a female assigned to extensive treatment?
b. What is the probability of being male or being assigned to no treatment?
c. What is the probability of being assigned to conventional treatment?

Question

Over the following three months, 300 convicted low-level drug offenders were assigned to different levels of treatment utilizing the methodology described above.
 $$\begin{array} { | l | l | l | l | l | } 
\hline & \text { Extensive treatment } & \text { Conventional treatment } & \text { No treatment } & \text { Total } \
\hline \text { Female } & 41 & 33 & 51 & 125 \
\hline \text { Male } & 59 & 67 & 49 & 175 \
\hline \text { Total } & 100 & 100 & 100 & 300 \
\hline
\end{array}$$ As the table indicates, we have a total of 300 inmates who were assigned to extensive treatment, conventional treatment, or no treatment according to the methodology presented in question 1. Of the 125 females, 51 have been assigned to no treatment, 33 to conventional treatment, and 41 to extensive treatment. The remaining 175 offenders were male, 49 who were assigned to no treatment, 67 to conventional treatment, and 59 to extensive treatment. 
a. If we selected one individual at random from the sample, what is the probability that a person is a female assigned to extensive treatment?
b. What is the probability of being male or being assigned to no treatment?
c. What is the probability of being assigned to conventional treatment?

Quizplus · Accepted Answer

The Answer is a. STEP 1: Calculate the probability of being female: P(female).
125/300 = 0.4167
The probability of being female is 0.4167, or about a 42% chance.
STEP 2: Calculate the probability of being assigned to an extensive treatment: P(extensive treatment), given being female.
P(extensive treatment|female) = 41/125 = 0.328
STEP 3: Multiply these probabilities together:
P(female) × P(extensive treatment|female) = 0.4167 × 0.328 = 0.1367, or about 13.67%
STEP 4: We can now interpret the probability of selecting a female who was assigned to extensive treatment. The chance that a person from the overall sample is female and is assigned to intensive treatment is 0.1367, or 13.67%.
b. STEP 1: Figure out all necessary probabilities. In this case, P(male) and P(no treatment).
P(male) = 175/300 = 0.5833
P(no treatment) = 100/300 = 0.3333
P(male and no treatment) = 0.5833 × 0.28 = 0.1633
STEP 2: Plug these probabilities into the formula P(A or B) = P(A) + P(B) - P(A and B|A)
P(male or no treatment) = P(male) + P(no treatment) - P(male and treatment|male)
P(male or no treatment) = 0.5933 + 0.3333 - 0.1633 = 0.7633, or 76.33%
STEP 3: Now we can interpret our findings. The chance of selecting a person being male or assigned to no treatment is 0.7633, or 76.33%.
c. Following the same steps above, but using the marginal totals, we see that the total number of those in convention treatment is 100, compared to the total number of subjects, which is 300. Thus, 100/300 = 0.667, or 66.7% chance of being selected in conventional treatment.

Over the Following Three Months, 300 Convicted Low-Level Drug Offenders