Selecting a letter that is a vowel from the letters of a word where there are repeating letters

A single letter is selected at random from the word “PROBABILITY”. The probability and odds that it is a vowel is:
 s~8 p~1 «Set~8 Problem~2 » s~9 p~1

Solution

Experiment : Selecting a letter from the letters of the word "PROBABILITY"

In the word "PROBABILITY"

Number of letters = 11 {P, R, O, B, A, B, I, L, I, T, Y}
Number of unique letters = 9 {P, R, O, B, A, I, L, T, Y} [B, I repeat]

Where there are repetitions, drawing any one of the repeated letters would mean the same. Thus only the unique letters are considered for finding the number of possible choices.

Total Number of Possible Choices

= Number of ways in which a unique letter can be drawn from the total 9 unique letters
⇒ n = 9C1
=
 9 1
= 9

Let,

• A : The event of the letter selected being a vowel

• For Event "A"

Number of vowels in the unique letters of the word "PROBABILITY" = 3 {O, A, I}

FavorableUnfavorable
VowelsOthersTotal
Available369
To Choose101
Choices3C16C09C1

Number of favorable/favourable choices

= Number of ways in which one letter which is a vowel can be selected from the total 3 vowels
⇒ mA = 3C1
=
 3 1
= 3

Probability of the letter drawn being a vowel

⇒ Probability of occurrence of Event 'A'

=
 Number of favourable/favorable choices for the Event Total number of possible choices for the Experiment
⇒ P(A) =
 mA n
=
 3 9

Odds

Number of unfavorable choices
 ⇒ mAc n - mA = Total number of possible choices - Number of favorable choices = = 9 - 3 = 6
in favor/favour
Odds in favor/favour of the letter selected being a vowel

⇒ Odds in favor/favour of Event 'A'

 = Number of favourable choices : Number of unfavorable choices = m : mAc = 3 : 6 = 1 : 2

against
Odds against the letter selected being a vowel

⇒ Odds against Event 'A'

 = Number of unfavourable choices : Number of favorable choices = mAc : m = 6 : 3 = 2 : 1