Bank Math Master

Question: n(A\B) + n(A ∩ B) = ?

Solution:
We know that,
A\B = {x : x ∈ A and x ∉ B} 

Example: A = {a, b, c} and B = {c, d}, 
then A ∩ B = {a, b, c} ∩ {c, d} = {c}
and A\B = {a, b, c} \ {c, d} = {a, b}

We can say that,
n(A\B) = n(A) - n(A ∩ B)

Now,
n(A\B) + n(A ∩ B) = n(A) - n(A ∩ B) + n(A ∩ B) = n(A)

Question: If A = {a, b, c, d, e, f, g} and B = {d, e, f, g}, then A - B = ?

Solution:
A - B = {a, b, c, d, e, f, g} - {d, e, f, g}
= {a, b, c}

Question: If B = {1, 2, 3, 4} and C = {x, y, z}, then B ∪ C = ?

Solution:
B ∪ C = {1, 2, 3, 4} ∪ {x, y, z}
= {1, 2, 3, 4, x, y, z}

Question: If n(A) = 39, n(B) = 23 and n(A ∩ B) = 19, then n(A ∪ B) = ?

Solution:
We know that,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
= 39 + 23 - 19
= 62 - 19
= 43

Question: Three dice are thrown together. Find the probability of getting a total of at least 6?

Solution:
The total possible ways of sum of 3 dice less than, 6
(1 + 1 + 1), (1 + 1 + 2), (1 + 1 + 3), (1 + 2 + 1), (1 + 2 + 2), (1 + 3 + 1), (2 + 1 + 1), (2 + 1 + 2), (2 + 2 + 1), (3 + 1 + 1)

Total 10 ways.
Total Sample space = 6³ = 216

Probability of getting the sum less than 6 = 10/216
Probability of getting the sum at least 6 = 1 - (10/216)
= (216 - 10)/216
= 206/216
= 103/108

Question: A fair coin is tossed 4 times. What is the probability of getting exactly 3 heads?

Solution:
For 4 tosses, the total number of possible outcomes is = 2⁴ = 16
The number of ways to choose 3 Heads out of 4 tosses is = ⁴C₃ = 4

∴ Probability of getting exactly 3 heads = 4/16 = 1/4

Question: A bag contains 4 green balls and 3 yellow balls. If two balls are drawn without replacement, what is the probability that both are yellow?

Solution:
Total balls = 4 green + 3 yellow = 7 balls.
Probability that the first ball is yellow = 3/7

After removing 1 yellow ball, we have:
Remaining yellow balls = 2
Total remaining balls = 6
So, the probability that the second ball is yellow = 2/6 = 1/3

∴ Total probability (both yellow) = 3/7 × 1/3 = 1/7

Question: If you toss a coin twice, what is the probability that you will get heads the second time?

Solution:
When we toss a coin the sample outcomes are = {HH, HT, TH, TT}
Total number of outcome = 4

Heads in the second time {HH, TH} = 2

∴ Probability that I will get heads the second time = 2/4 = 1/2 = (1 × 100)/2 % = 50%

Question: Two unbiased coins are tossed. What is the probability of getting at most one head?

Solution:
Total cases = {HH, HT, TH, TT} = 4
Favorable cases = {TT, HT, TH} = 3

∴ Required Probability = 3/4

Question: What is the mean of the range, mode and median of the data given below? 
5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4

Solution:
Given data = 5, 10, 3, 6, 4, 8, 9, 3, 15, 2, 9, 4, 19, 11, 4
Arranging in ascending order = 2, 3, 3, 4, 4, 4, 5, 6, 8, 9, 9, 10, 11, 15, 19
Here,
Most frequent data is 4
So, Mode = 4

Total terms in the given data, (n) = 15 (It is odd)
∴ Median = {(n + 1)/2}^th term
= {(15 + 1)/2}^th term
= (8)^th term
= 6

Now,
Range = (Maximum value - Minimum value) + 1
= (19 - 2) + 1
= 18

∴ Mean of Range, Mode and Median = (Range + Mode + Median)/3
= (18 + 4 + 6)/3 = 28/3 = 9.33

Question: Which value of y will satisfy the given inequality, 2(y - 3) ≥ 3y - 4 ?

Solution:
Given,
2(y - 3) ≥ 3y - 4
⇒ 2y - 6 ≥ 3y - 4
⇒ 2y - 3y ≥ - 4 + 6
⇒ - y ≥ 2
⇒ y ≤ - 2

Question: Solve the inequality 2 ≤ - 4 - 3x < 17

Solution:
2 ≤ - 4 - 3x < 17
⇒ 2 + 4 ≤ - 4 - 3x + 4 < 17 + 4
⇒ 6 ≤ - 3x < 21
⇒ - 6 ≥ 3x > - 21
⇒ - 6/3 ≥ 3x/3 > - 21/3
⇒ - 2 ≥ x > - 7
∴ - 7 < x ≤ - 2

Question: If x ≥ 7 and y ≤ 4 which of the following must be true?

Solution:
Given that,
x ≥ 7
and y ≤ 4
⇒ - y ≥ - 4

Now,
x - y ≥ 7 - 4
∴ x - y ≥ 3

Question: Solve the inequality |x - 2| < 5

Solution:
|x - 2| < 5
⇒ - 5 < x - 2 < 5
⇒ - 5 + 2 < x - 2 + 2 < 5 + 2
⇒ - 3 < x < 7

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

Solution:
1 - x² ≥ 0
⇒ - x² ≥ - 1
⇒ x² ≤ 1
⇒ x² ≤ 1²
∴ - 1 ≤ x ≤ 1

Question: Find out the missing number : 52, 64, 76, ____ , 100

Solution:
52 + 12 = 64
64 + 12 = 76
76 + 12 = 88
88 + 12 = 100

Question: What is the 9^th term of the sequence : - 2, - 4, - 6, ............................ , - 100?

Solution:
Here,
- 4 - (- 2) = - 4 + 2 = - 2
- 6 - (- 4) = - 6 + 4 = - 2
∴ d = - 2
a = - 2
n = 9

∴ The 9^th term of the sequence = a + (n - 1)d
= - 2 + (9 - 1) (- 2)
= - 2 + 8 (- 2)
= - 2 - 16
= - 18

Question: Find out the missing number in the following series. 256, 16, 4, ____?

Solution:
√256 = 16
√16 = 4
√4 = 2

So, missing number is = √4 = 2

Question: In a series of 5 consecutive odd numbers if 13 is the 5^th number, what is the 3^rd number in the series?

Solution:
5^th odd number = 13
4^th odd number = 11
3^rd odd number = 9
2^nd odd number = 7
1^st odd number = 5

Question: The next term of the series: 25, 49, 81, ____ is

Solution:
Given: 25, 49, 81, ____
The series is: 5², 7², 9², 11²
So, the next term is 11² = 121