Conditional Probabilty

1. Question: There are 50 students enrolled in the second year of a business degree program. During this semester, the students have to take some elective courses. 18 students are enrolled in psychology, 27 students are enrolled in geography, and 10 students are enrolled in both psychology and geography. A random student is to be selected. What is the probability that this student is enrolled in psychology or geography?

   P(Psychology) = 18/50

   P(Geology) = 27/50

   P(PnG) = 10/50

   10 in both so 8 in only Psycholgoy and 17 in only geology

   10+8+17 = 35

   P(PUG) = 35/50

   Formula

   P(PUG) = P(P) + P(G) - P(PnG)

2. At a local language school, 40% of the students are learning Spanish, 20% of the students are learning German, and 8% of the students are learning both Spanish and German. What is the probability that a randomly selected student is learning Spanish or German? May be useful to see a Venn Diagram out of 100 people.

P(S) = 40/100

P(G) = 20/100

P(SnG) = 8/100

P(PUG) = P(P) + P(G) - P(PnG)

= 40/100 + 20/100 8/100 = 52/100

P(A|B) = probability that event A will happen GIVEN that Event B has happened.

Example:

Researchers randomly assigned 72 chronic users of cocaine into three groups: desipramine (antidepressant), lithium (standard treatment for cocaine) and placebo. Results of the study are summarized below.

	relapse	no relapse	Total
desipramine	10	14	24
lithium	18	5	23
placebo	20	5	25
Total	48	24	72

If I selected a patient at random from this study, what is the size of the Sample Space S?

It would be 72 people I could pick.

P(replapsed) = 48/72

P(desipramine) = 24/72

p(replapsed and desipramine) = 10/72

FInd the probability that a patient was on desipramine given that they relapsed

find P(desipramine|relapsed)

P(desipramine|relapsed) = 10/48

P(relapsed|desipramine) = 10/24

10/24 = (10/72)/(24/72)

= P(relapsed ∩ desipramine)/P(desipramine)

So,

P(A|B)=P(A ∩ B)/P(B) in general