← Back to topic
Vidaara.orgClass 12 · Mathematics
CodeVID-M12-13-BAY-01
Total Probability & Bayes’ Theorem — Assignment
Chapter: Probability
Topic: Total Probability & Bayes’ Theorem
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
The law of total probability gives $P(A)=$
  • A.$\sum_i P(E_i)P(A\mid E_i)$
  • B.$P(A)P(B)$
  • C.$\max_i P(A\mid E_i)$
  • D.$\sum_i P(E_i)$
2.
Bayes' theorem computes:
  • A.$P(A\mid E_k)$
  • B.$P(E_k\mid A)$
  • C.$P(A\cap E_k)$
  • D.$P(E_k)$
3.
The posterior probability is computed:
  • A.before evidence
  • B.after observing $A$
  • C.ignoring $A$
  • D.from priors only
4.
In the two-bag example, $P(\text{red})=$
  • A.$\tfrac25$
  • B.$\tfrac35$
  • C.$\tfrac12$
  • D.$\tfrac{3}{10}$
5.
In Bayes' theorem the events $E_i$ must be:
  • A.independent
  • B.mutually exclusive and exhaustive
  • C.equal
  • D.continuous
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
If $P(E_1)=0.4,\ P(E_2)=0.6,\ P(A\mid E_1)=0.5,\ P(A\mid E_2)=0.5$, find $P(A)$.
7.
State the difference between prior and posterior probability.
8.
If $P(E_1)=P(E_2)=\tfrac12,\ P(A\mid E_1)=0.2,\ P(A\mid E_2)=0.8$, find $P(A)$.
9.
Bag I has $3$R, $2$B; Bag II has $1$R, $4$B. A bag is chosen at random and a red ball drawn. Find $P(\text{red})$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
For the two bags above, find $P(\text{Bag I}\mid \text{red})$.
11.
A factory: machine A makes $60\%$, B $40\%$; defective rates $2\%$ and $3\%$. Find $P(\text{defective})$.
12.
For that factory, find $P(A\mid \text{defective})$.
13.
If $P(E_1)=0.5,P(E_2)=0.3,P(E_3)=0.2$ and $P(A\mid E_i)=0.1,0.2,0.3$, find $P(A)$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Machines A, B, C produce $25\%,35\%,40\%$ of items with defective rates $5\%,4\%,2\%$. An item is defective; find the probability it was made by C.
15.
A disease affects $1\%$ of a population. A test is $99\%$ sensitive and gives a $2\%$ false positive. Given a positive test, find the probability of disease.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $\sum_i P(E_i)P(A\mid E_i)$
  2. (B) $P(E_k\mid A)$
  3. (B) after observing $A$
  4. (A) $\tfrac25$
  5. (B) mutually exclusive and exhaustive
Section B — Short Answer (2 marks)
  1. $0.5$.
  2. Prior is before evidence; posterior is after observing the event.
  3. $0.5$.
  4. $\tfrac25$.
Section C — Short Answer (3 marks)
  1. $\tfrac34$.
  2. $0.024$.
  3. $0.5$.
  4. $0.17$.
Section D — Long Answer (5 marks)
  1. $\tfrac{16}{69}\approx0.232$.
  2. $\tfrac13\approx0.333$.
Generated by Vidaara.org · Assignment VID-M12-13-BAY-01 · vidaara.org