Two students are registered for the same class and attend independently of each
other, student A 70% of the time and student B 60% of the time. The teacher
remembers that on a given day at least one of them is in class. What is the
probability that student A was in class that day?
This is conditional probability. The probability that A was in class, given
that A or B was in class that day.
The formula is
P(X and Y)
P(X|Y) = ------------
P(Y)
where X = A, Y = (A or B)
P(A and (A or B)) P(A)
P(A|A or B) = ----------------- = ---------
P(A or B) P(A or B)
We know P(A) = .70. We need P(A or B)
The complement of 'at least 1' is 'none at all'.
student A attends 70% of the time.
Therefore A is absent 30% of the time. Probability .3
Student B attends 60% of the time. Probability .6
Therefore B is absent 40% of the time. Probability .4
They are both absent (.3)(.6) = .18 or 18% of the time.
So 1 or the other is present 82% of the time. Probability =.82.
P(A and (A or B)) P(A) .3 30 15
P(A|A or B) = ----------------- = --------- = ------ = ---- = ----
P(A or B) P(A or B) .82 82 41
Probability about .366 or 36.6%of the time.
Edwin