Question

Question1

b) The English alphabet contains 21 consonants and 5 vowels. How many strings of

six lower case letters of the English alphabet contain:i) Exactly one vowel?

i) Exactly 2 vowels

III) At least 1 vowel

IV) At least 2 vowels)

c).If A and B are independent events, show that A and B are also independent. Are A And B independent?

d) Find Population Mean, median, mode and Sample Standard Deviation for the following

data set: 5, 10, 15, 20, 25, 30

Answer 1

Part b)

i) Exactly 2 vowels

There are 26 letters in a total of which 21 are consonants and 5 are vowels. We are interested in strings contains 6 letters. 

Position vowel: 6 ways (as there are 6 letters in the string) 

Vowel: 5 ways

Second letter: 21 ways (needs to a consonant)

Third letter: 21 ways (needs to be a consonant)

Fourth letter: 21 ways (needs to be a consonant.)

Fifth letter: 21 ways (needs to be a consonant)

Sixth letter: 21 ways (needs to be a consonant) 

Use the product rule: 

$6* 5*21*21*21*21*21= 6*5*21^5= 122,523,030$

Thus there are 122,523,030 strings containing exactly one vowel. 

III) At least 1 vowel

There are 26 letters in a total of which 21 are consonants and 5 are vowels. We are interested in strings contains 6 letters. 

There are 26 possible letters for each letter in the string. By the product rule: 

Number of strings = $26* 26* 26* 26* 26 *26 = 26^6 = 308,915,776$

When the string contains no vowels, then there are 21 possible letters for each letter in the string. By the product rule: 

Number of strings with no vowels = $21 *21 *21* 21* 21* 21 = 21^6 = 85,766,121$

Strings that do not have any vowels will have at least one vowel. 

Number of strings with at least one vowel = Number of strings — Number of strings with no vowels = $26^6 - 21^6 = 308,915,776 - 85,766,121 = 223,149,655$

IV) At least 2 vowels

Number of strings with at least two Nrowels

= Number of strings — Number of strings with no Nrowels — Number of strings with at least one vowel

=$26^6 — 21^6 — 6 * 5 * 21^5$

$= 308,915,776— 85,766,121 — 122,523,030 = 100,626,625$

c) The events A and B are independent, so, $P(A ∩ B) = P(A) P(B)$ .

$A = ( A ∩ B) ∪ (A ∩ B’).$

Also, $P(A) = P[(A ∩ B) ∪ (A ∩ B’)].$

or,$P(A) = P(A ∩ B) + P(A ∩ B’).$

or, $P(A) = P(A) P(B) + P(A ∩ B’)$

or, $P(A n B’) = P(A) − P(A) P(B) = P(A) (1 – P(B)) = P(A) P(B’)$

d)Mean

$\frac{ 5+ 10+ 15+ 20+ 25+ 30}{6}=17.5$

median

15, 20

$\frac{ 15+ 20}{2}=17.5$

mode

None

Sample Standard

$s= \sqrt{ \frac{ \sum(x_i- \bar{x})^2}{N-1}}= \sqrt{ \frac{437.5}{6-1}}= \sqrt{87.5}=9.35$