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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-13-CT
Probability — Full Chapter Test
Chapter: Probability
Topic: Complete chapter — all topics
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • This is a full-length test covering the whole chapter — every topic is included.
  • 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.
$P(A\mid B)$ equals:
  • A.$\dfrac{P(A\cap B)}{P(B)}$
  • B.$P(A)P(B)$
  • C.$\dfrac{P(B)}{P(A)}$
  • D.$P(A)+P(B)$
2.
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.$\sum_i P(E_i)$
3.
For a valid distribution, $\sum p_i=$
  • A.$0$
  • B.$1$
  • C.$n$
  • D.$\mu$
4.
$A,B$ are independent means $P(A\cap B)=$
  • A.$0$
  • B.$P(A)P(B)$
  • C.$P(A)+P(B)$
  • D.$P(A\mid B)$
5.
Bayes' theorem computes:
  • A.$P(A\mid E_k)$
  • B.$P(E_k\mid A)$
  • C.$P(A\cap E_k)$
  • D.$P(E_k)$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
If $P(A)=0.5,\ P(B)=0.4,\ P(A\cap B)=0.2$, are $A,B$ independent?
7.
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 $P(A)$.
8.
For a valid probability distribution, what is $\sum p_i$?
9.
Two cards are drawn without replacement. Find $P(\text{both kings})$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
A die is rolled. Find $P(\text{even}\mid \text{number}>2)$.
11.
For the two bags above, find $P(\text{Bag I}\mid \text{red})$.
12.
Find $E(X)$ for $X:1,2,3$ with $p:\tfrac16,\tfrac26,\tfrac36$.
13.
From a bag of $5$ red and $3$ black balls, two are drawn without replacement. Find $P(\text{both red})$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
A family has two children. Find the probability that both are girls given that at least one is a girl.
15.
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.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $\dfrac{P(A\cap B)}{P(B)}$
  2. (A) $\sum_i P(E_i)P(A\mid E_i)$
  3. (B) $1$
  4. (B) $P(A)P(B)$
  5. (B) $P(E_k\mid A)$
Section B — Short Answer (2 marks)
  1. Yes.
  2. $0.5$.
  3. $1$.
  4. $\tfrac{1}{221}$.
Section C — Short Answer (3 marks)
  1. $\tfrac12$.
  2. $\tfrac34$.
  3. $\tfrac73$.
  4. $\tfrac{5}{14}$.
Section D — Long Answer (5 marks)
  1. $\tfrac13$.
  2. $\tfrac{16}{69}\approx0.232$.
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