BANK & INSURANCE (PROBABILITY) PART 1

Total Questions: 45

31. Two dice are thrown simultaneously. What is the probability of getting the face numbers are same?

Correct Answer: (a) 1/6  
Solution:

In a simultaneous throw of two dice, we have n(s) = 6 × 6 = 36
Let E = event of getting two numbers are same.
Then E = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
therefore, n(E) = 6
And P(E) = p(getting two numbers are same)
P(E) = n(E)/n(s) = 6/36 = 1/6
Hence the answer is 1/6

32. A bag contains 6 black and 2 white balls and another bag contains 6 black and 8 white balls. If one of the bag is selected at random and two balls are selected at random from the bag thus selected, then what is the probability that the two balls selected are of different colours?

Correct Answer: (e) None of these
Solution:Probability of selecting any bag = 1/2
Probability of selecting a black and a white ball from
1st bag = (⁶C₁ × ²C₁)/⁸C₂ = 3/7
Probability of selecting a black and a white ball from
2nd bag (⁶C₁ × ⁸C₁)/¹⁴C₂ = 24/91
Required probability = P (1st bag) P (a black ball and
a white ball from, 1st bag) + P (2nd bag) P (a black ball
and a white ball from 2nd bag)
= 1/2 × 3/7 + 1/2 × 24/91 = 63/182

33. In a test, the probability of Raman scoring a 100/100 is 1/5. In a school exam of 6 papers, what is the probability (in approximate percentage) of Raman scoring a 100/100 in at least 2 papers?

Correct Answer: (b) 34%  
Solution:

Probability that Raman scores 100/100 in exactly 2 papers = ⁶C₂ × (1/5)² (4/5)⁴ = 0.24576
Probability that Raman scores 100/100 in exactly 3 papers = ⁶C₃ × (1/5)³ (4/5)³ = 0.08192
Probability that Raman scores 100/100 in exactly 4 papers = ⁶C₄ × (1/5)⁴ (4/5)² = 0.01536
Probability that Raman scores 100/100 in exactly 5 papers = ⁶C₅ × (1/5)⁵ (4/5) = 0.001536
Probability that Raman scores 100/100 in exactly 6 papers = ⁶C₆ × (1/5)⁶ = 0.000064

Required probability = 0.24576 + 0.08192 + 0.01536 + 0.001536 + 0.000064 = 0.34464 ~ 34%

34. There are two bags, one of which contains 6 black and 8 white balls, while the other contains 8 black and 6 white balls. A dice is cast. If the face 2 or 6 turns up, a ball is taken from the first bag and if any other face turns up a ball is chosen from the second bag. The probability of choosing a black ball is:

Correct Answer: (b) 11/21  
Solution:Bag 1 contains 6 black and 8 white balls.
Bag 2 contains 8 black and 6 white balls.
P (Bag 1 is chosen) = 2/6 = 1/3
P (Bag 2 is chosen) = 4/6 = 2/3
P (black ball is drawn from bag 1) = 6/14 = 3/7
P (black ball is drawn from bag 2) = 8/14 = 4/7
P (black ball) = 3/7 × 1/3 + 4/7 × 2/3 = 1/7 + 8/21 = 11/21

35. A bag contains 3 blue and 4 black pen and another bag contains 4 blue and 3 black pen. A dice is cast and if the face 2 or 5 turns up, a pen is taken from the first bag and if any other face turns up, a pen is taken from the second bag. What would have been the increase/decrease in probability of drawing a black pen if the first bag was selected when the dice’s face turned out to be 1,2 or 5, and bag 2 was selected when any other number.

Correct Answer: (c) 1/42 increase  
Solution:The probability of selection of first bag = 1/3.
The probability of selecting a black pen from the first bag = 4/7.
The probability of selection of first bag = 2/3.
Probability of selecting a black pen from the second bag = 3/7.
Probability of selecting a black pen = 4/21 + 6/21 = 10/21.
In the second case, probability of selecting both bags become 1/2.
Therefore, change = 4/14 + 3/14 − 10/21 = 1/42 increase.

36. A box contains 3 red, 4 yellow and (x + 1) green balls. If two balls are taken out, then the probability that both the balls being green is 5/33. Find the value of x?

Correct Answer: (b) 4  
Solution:

(x + 1)C₂ / (8 + x)C₂ = 5/33
5 × (8 + x) × (7 + x) = 33 × (x + 1) × x
5 × (56 + 8x + 7x + x²) = 33x² + 33x
280 + 75x + 5x² = 33x² + 33x
28x² − 42x − 280 = 0
2x² − 3x − 20 = 0
x = 4, −5/2
So, the value of x = 4

37. A box contains three red, four yellow and one green balls. If three balls are drawn at random, what is the probability that two of them are red and one green?

Correct Answer: (c) 3/56
Solution:Required probability = (³C₂ × ¹C₁)/⁸C₃ = 3/56

38. A bag contains 5 red and x yellow balls. If two balls are drawn at random, then the probability of that balls being red is 5/33. Find the value of x.

Correct Answer: (c) 7
Solution:

⁵C₂/(5 + x)C₂ = 5/33
(5 + x)(4 + x) = 4 × 33
20 + 5x + 4x + x² = 132
x² + 9x − 112 = 0
x² + 16x − 7x − 112 = 0
x(x + 16) − 7(x + 16) = 0
x = 7

39. A bag contains 12 blue balls, 7 white balls and 6 yellow balls. If 2 balls are drawn at random, then find the probability of getting all the balls are same in colour?

Correct Answer: (b) 17/50  
Solution:

Required probability = (¹²C₂ + ⁷C₂ + ⁶C₂)/²⁵C₂
= (66 + 21 + 15)/300 = 102/300 = 17/50

40. A bag contains 3 red, 5 yellow and 4 blue balls. If three balls are drawn at random, then find the probability of that balls are yellow?

Correct Answer: (a) 1/22  
Solution:Required probability = ⁵C₃/¹²C₃
= 5 × 4 × 3 / 12 × 11 × 10 = 1/22