Question

3. An experiment consists of randomly rearranging the 10 letters of the word
QUARANTINE
into a sequence of 10 letters, where all possible orders of these 10 letters are equally likely. Find the probability of each of the following events:
(1) 2 the first three letters include no A’s;
(2) 3 the first three letters or the last three letters (or both) include no A’s;
(3) 2 the fourth letter is the first A;
(4) 3 the first letter and the last letter are the same;
(5) 2 the word ‘QUARANTINE’ is obtained;
(6)3 the sequence contains the word ‘RAN’

Answer 1

Given that:

An experiment consists of randomly rearranging the 10 letters of the word QUARANTINE into a sequence of 10 letters, where all possible orders of these 10 letters are equally likely.

The word is QUARANTINEE

Q=1,U=1,A=2,R=1,N=2,T=1,I=1,E=1

(1) the first three letters include no A’s;

Let Event A = The first three letters include no A's

Favourable outcome: Two A's can fill column 4 to 10.And rest letter can occupy any remaining column

Note: Number of arrangements of n letter words in which one letter repeat r1 times another letter repeat r2 times and ......nth letter repeat rn times is

(2)  the first three letters or the last three letters (or both) include no A’s;

Let event B =last three letters don't include A's

We have to find

Now,outcome favorable to Event B -Two A's can occupy any column from 1 to 7 and the rest letter are occupied remaining space

Outcome variable to event

i.e No A's occupy either the first 3 letters or last 3 letter it means both A,s have to occupy the column 4 to 7.And rest letter occupy the remaing partition

(3) the fourth letter is the first A;

Let Event C= The fourth letter is the first A

Favourable outcome: Second A's can occupy any position from 5 to 10 column and rest letter occupy any remaining columns

(4) the first letter and the last letter are the same;

Let Event D ; First letter and last letters are same

Favourable outcome:Remaing letters can occupy any position from 2nd tp 9th column

(5) the word ‘QUARANTINE’ is obtained;

Word 'QUARANTINE' is obtained

(6) the sequence contains the word ‘RAN’.

Let Event E = sequance contains the word RAN

Favorable outcomes: The sequences 'RAN' can occupy positions (1,2,3),(2,3,4) (3,4,5) ,....(8,9,10)

Total 8 ways

And the remaining letters can occupy any remaining position