Answer to Question #177093 in Statistics and Probability for kojo

Question

1.     Of the students in the college, 60% of the students reside in the hostel and 40% of the students are day scholars. Previous year result reports that 30% of all students who stay in the hostel scored A Grade and 20% of day scholars scored A grade. At the end of the year, one student is chosen at random and found that he/she has an A grade. What is the probability that the student is a hostlier? 

2.      Two players A and B are competing at a trivia quiz game involving a series of questions. On any individual question, the probabilities that A and B give the correct answer areαandβrespectively, for all questions, with outcomes for different questions being independent. The game finishes when a player wins by answering a question correctly. Compute the probability that A wins if

a)     A answers the first question,

b)     B answers the first question.

Answer 1

1. Let A denote A grade.

H denote from hostel.

D denote day scholar.

P(A|H)=0.3

P(A|D)=0.2

P(H)=0.6

P(D)=0.4

$P(H|A)= \frac{P(A|H) \times P(H)}{P(A|H) \times P(H)+P(A|D) \times P(D)} \\ P(H|A)= \frac{0.3 \times 0.6}{0.3 \times 0.6 + 0.2 \times 0.4} \\ = \frac{0.18}{0.18 + 0.08} \\ = \frac{0.18}{0.29} \\ = \frac{9}{13}$

2. Let event A - A answers the first question;

event F - game ends after the first question;

event W - A wins.

To find:

P(W|A′)

P(F|A)=P [A answers first question correctly] = α

P(F′|A)=1−α

$P(W|A \cup F)=1 \\ P(W|A \cup F′)=P(W/A′) \\ P(W|A)=P(W|A \cup F)P(F/A)+P(W|A \cup F′)P(F′|A) \\ P(W|A)=(1 \times α)+(P(W|A′) \times (1−α))=α+P(W|A′)(1−α)$

P(F|A′)=P [B answers first question correctly] = β

P(F′|A)=1−β

$P(W|A′ \cup F)=0 \\ P(W|A′ \cup F′)=P(W|A) \\ P(W|A′)=(0 \times β)+(P(W|A) \times (1−β))=P(W|A)(1−β) \\ P(W|A′)= \frac{(1−β)α}{1−(1−α)(1−β)}$