Question

In the senior year of a high school graduating class of 80 students, 34 studied mathematics, 58 studied psychology, 44 studied history, 16 studied both mathematics and history, 23 studied both mathematics and psychology, 7 studied history but neither mathematics nor psychology, 8 studied all three subjects, and 4 did not take any of the three. If a student who studied mathematics is selected, what is the probability that the student has also studied psychology? [The answer should be a number rounded to five decimal places, don't use symbols such as %]

Answer 1

Let us denote the event

M : a randomly selected student studied Mathematics

P : a randomly selected student studied psychology

H : a randomly selected student studied history

Given

$n(M) = 34, n(P) = 58, n(H) = 44, n(M\cap H) = 16, n(M\cap P) = 23, n(M^c\cap P^c \cap H) = 7, n(M\cap P \cap H) = 8, n(M^c\cap P^c \cap H^c) = 4$

and n : total number of observations

If a student who studied mathematics is selected,

The probability that the student has also studied psychology

$=P(P|M)$

$=\frac{P(P\cap M)}{P(M)}$

$=\frac{\frac{n(P\cap M)}{n}}{\frac{n(M)}{n}}$

$=\frac{n(P\cap M)}{n(M)}$

$=\frac{23}{34}$

$\approx 0.67647$