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16. IEB NOV 2014 QUESTION 6

16.1 In a sample space [tex]\( S \)[/tex], the number of elements in [tex]\( S \)[/tex], [tex]\( n(S) = 30 \)[/tex], and there are two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] such that [tex]\( n(A) = 15 \)[/tex], [tex]\( n(B) = 20 \)[/tex] with [tex]\( n(A \cap B) = 6 \)[/tex].

16.1.1 Draw a Venn diagram to represent this situation.
(3) L2

16.1.2 Write down the value of [tex]\( n(A \cup B) \)[/tex].
(1) L1

16.1.3 An element is randomly selected from [tex]\( S \)[/tex]. Write down the probability that the element is in both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. That is, [tex]\( P(A \cap B) \)[/tex].
(1) L1

16.1.4 Showing all working, determine whether the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
(3) L3

16.2 Steve needs to set up a format for passwords on his website. He has decided on using letters from the alphabet (26 letters) followed by digits 0 to 9. Letters and digits can be repeated.

16.2.1 Calculate the number of passwords that can be created using 2 letters followed by 2 digits.
(2) L2

16.2.2 Steve thinks that he will need to cater for 3 million different passwords. He will stick with 2 letters but will need more digits. Determine the least number of digits he will need.
(1) L1



Answer :

GinnyAnswer
Let's work through the problem step-by-step.

### 16.1 Events in a Sample Space

Given:
- [tex]\( n(S) = 30 \)[/tex] (total number of elements in the sample space [tex]\( S \)[/tex])
- [tex]\( n(A) = 15 \)[/tex] (number of elements in event [tex]\( A \)[/tex])
- [tex]\( n(B) = 20 \)[/tex] (number of elements in event [tex]\( B \)[/tex])
- [tex]\( n(A \cap B) = 6 \)[/tex] (number of elements in both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex])

#### 16.1.1 Venn Diagram

To represent this situation with a Venn diagram:

1. Draw a rectangle to represent the sample space [tex]\( S \)[/tex] with [tex]\( n(S) = 30 \)[/tex].
2. Within [tex]\( S \)[/tex], draw two overlapping circles, one for event [tex]\( A \)[/tex] and the other for event [tex]\( B \)[/tex].
3. Label the circles with [tex]\( n(A) = 15 \)[/tex] and [tex]\( n(B) = 20 \)[/tex]. The overlapping area represents [tex]\( n(A \cap B) = 6 \)[/tex].

The Venn diagram helps visualize the relationships between the sets.

#### 16.1.2 Value of [tex]\( n(A \cup B) \)[/tex]

To find the value of [tex]\( n(A \cup B) \)[/tex], we use the principle of inclusion and exclusion:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]

Plugging in the given values:
[tex]\[ n(A \cup B) = 15 + 20 - 6 = 29 \][/tex]

So, [tex]\( n(A \cup B) = 29 \)[/tex].

#### 16.1.3 Probability [tex]\( P(A \cap B) \)[/tex]

The probability that a randomly selected element from [tex]\( S \)[/tex] is in both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[ P(A \cap B) = \frac{n(A \cap B)}{n(S)} \][/tex]

Plugging in the given values:
[tex]\[ P(A \cap B) = \frac{6}{30} = 0.2 \][/tex]

So, the probability [tex]\( P(A \cap B) \)[/tex] is [tex]\( 0.2 \)[/tex].

#### Determining Independence

Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if:
[tex]\[ P(A \cap B) = P(A) \times P(B) \][/tex]

First, find [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex]:
[tex]\[ P(A) = \frac{n(A)}{n(S)} = \frac{15}{30} = 0.5 \][/tex]
[tex]\[ P(B) = \frac{n(B)}{n(S)} = \frac{20}{30} = 0.6667 \][/tex]

Now, calculate:
[tex]\[ P(A) \times P(B) = 0.5 \times 0.6667 = 0.3333 \][/tex]

Since:
[tex]\[ P(A \cap B) = 0.2 \neq P(A) \times P(B) = 0.3333 \][/tex]

Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not independent.

### 16.2 Password Format

Given:
- Alphabet of 26 letters.
- Digits from 0 to 9 (10 digits in total).
- Letters and digits can repeat.

#### 16.2.1 Number of Passwords with 2 Letters Followed by 2 Digits

Each position can be occupied as follows:
- First letter: 26 choices
- Second letter: 26 choices
- First digit: 10 choices
- Second digit: 10 choices

Total number of passwords:
[tex]\[ 26^2 \times 10^2 = 676 \times 100 = 67600 \][/tex]

So, the number of passwords that can be created using 2 letters followed by 2 digits is 67,600.

#### 16.2.2 Determining the Least Number of Digits for 3 Million Passwords

Steve needs at least 3 million different passwords. Let [tex]\( d \)[/tex] be the number of digits needed. The total number of passwords is given by:
[tex]\[ 26^2 \times 10^d = 3,000,000 \][/tex]

Simplify to solve for [tex]\( d \)[/tex]:
[tex]\[ 676 \times 10^d = 3,000,000 \][/tex]
[tex]\[ 10^d = \frac{3,000,000}{676} \approx 4438.497 \][/tex]

To find the smallest integer [tex]\( d \)[/tex] such that [tex]\( 10^d \geq 4438.497 \)[/tex]:
[tex]\[ d \approx \lceil \log_{10}(4438.497) \rceil \approx 3.65 \][/tex]
[tex]\[ d = 4 \][/tex]

So, Steve will need at least 4 digits to have 3 million different passwords.

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