Bank Math Master

Question: A die is thrown three times and the sum of the three numbers is found to be 15. The probability that the first throw was a four is

Solution: 
6 + 6 + 3 = 15, number of ways = 3!/2! = 3
6 + 5 + 4 = 15,  number of ways = 3! = 6
5 + 5 +5 = 15,  number of ways = 1 
Total ways = 10 

If first throw was a four = (4, 5, 6), (4, 6, 5) = 2 ways 

Probability = 2/10 = 1/5 

Question: An unbiased die is tossed.Find the probability of getting a multiple of 3.

Solution: 
Here S = {1, 2, 3, 4, 5, 6}
Let E be the event of getting the multiple of 3
Then,
E = {3,6}
P(E) = n(E)/n(S)
= 2/6
= 1/3

Question: Solve Ιx - 2Ι < 3

Solution:
Given
Ιx - 2Ι < 3
⇒ - 3 < x - 2 < 3
⇒ - 3 + 2 < x - 2 + 2 < 3 + 2
⇒ - 1 < x < 5

Question: Find the number that should be placed in the gap of the series : 64, 80, 96, _______, 128

Solution: 
64 + 16 = 80
80 + 16 = 96 
96 + 16 = 112 
112 + 16 = 128 

Question: If -1 < p < 0, then which of the following is the biggest? 

Solution: 
let, p = -1/2 

p/2 = -1/(2 × 2) = -1/4
p² = (-1/2)2 = 1/4
1/p² = 4 
p + 1 = 1/2

Question: A committee of 5 members is to be formed by selecting out of 6 men and 7 women. What is the probability that the committee has exactly 2 men and 3 women?

Solution:
Total member = 6 + 7 = 13
2 men can be selected out of 6 men in  ⁶C₂ ways
3 women can be selected out of 7 women in ⁷C₃ ways
Required number of ways = ⁶C₂ × ⁷C₃ = 15 × 35 = 525

The total number of ways to make committee with all members = ¹³C₅ = 1287

∴ The probability that the committee has exactly 2 men and 3 women = 525/1287
= 175/429

Question: m, n, o are natural numbers. If m> n> o then which of the following is not true?

Solution: let, m = 4, n = 3, o = 2
a) mno = 4 × 3 × 2 = 24>0
b) mn - o = 4 × 3 - 2 = 12 - 2 = 10 > 0
c) n - mo = 3 - 4 × 2 = 3 - 8 = -5 < 0

Question: Which of the following describes all values of x for which 1 - x² ≥ 0?

Solution: 
1 - x² ≥ 0
⇒ 1 ≥  x²
⇒ x² ≤ 1
⇒ √(x²) ≤ √1
⇒ |x| ≤ 1

We must consider that x can be either positive or negative because the variable x is inside the absolute value sign. Therefore, we’ll need to solve the inequality twice.

When x is positive:
|x| ≤ 1
= x ≤ 1

When x is negative:
|x| ≤ 1
= - x ≤ 1
= x ≥ - 1

We combine the two resulting inequalities to get:
-1 ≤ x ≤ 1

Question: If a > 7 and b > - 3 then which of the following is true?

Solution:
Given, 
a > 7 and b > - 3
⇒ ab > 7 × (- 3)
⇒ ab > - 21

Question: 0, 2, 6, 8, 16, 30, 54,?  

Solution: 
0 + 2 + 6 = 8
2 + 6 + 8 = 16
6 + 8 + 16 = 30
8 + 16 + 30 = 54

16 + 30 + 54 = 100

Question: Three unbiased coins are tossed. What is the probability of getting at least two heads?

Solution:
The events when three unbiased coins are tossed = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Total number of events 8

The events of getting at least two heads {HHH, HHT, HTH, THH}

Number of expected events = 4

∴ The probability of getting at least two heads is = 4/8 = 1/2