Question

The probability that a random student at Elmville College is a freshman is 0.3; a sophomore, 0.25; and a junior or senior, 0.45. The probability that a freshman will major in engineering is 0.15; a sophomore, 0.2; and a junior or senior, 0.3. What is the probability that a random student at this college majors in engineering?

Answer 1

Let $L_1$ denote the event that a student is a freshman, $L_2$ a sophomore, $L_3$ a junior, and $L_4$ a senior.

Let $E$ denote the event that a student majors in engineering. By the law of total probability,

$\mathbb P(E)=\mathbb P(E\cap L_1)+\mathbb P(E\cap L_2)+\mathbb P(E\cap L_3)+\mathbb P(E\cap L_4)$

By the definition of conditional probability, we can expand each of these intersection probabilities to get

$\mathbb P(E)=\mathbb P(E\mid L_1)\mathbb P(L_1)+\mathbb P(E\mid L_2)\mathbb P(L_2)+\mathbb P(E\mid L_3)\mathbb P(L_3)+\mathbb P(E\mid L_4)\mathbb P(L_4)$

$\mathbb P(E)=0.15\cdot0.3+0.2\cdot0.25+2(0.3\cdot0.45)=0.365$