Question

A letter is taken out at random from the word ASSISTANT and another from STATISTICS. The probability that they are the same latter is
(A) $\dfrac{{13}}{{90}}$
(B) $\dfrac{{17}}{{90}}$
(C) $\dfrac{{19}}{{90}}$
(D) $\dfrac{{15}}{{90}}$

Answer 1

Hint: For this type of probability question how many total numbers of alphabet are present in each word and what is the frequency of each single alphabet. Then find the probability of each one.
Now for this question we have to choose a common alphabet and the probability of the same alphabet from both latter gets multiplied and added to the next same alphabet probability and so on.

Complete step-by-step answer:
First letter is ASSISTANT so to solve question we have to find frequency of each alphabet that is given below
Frequency of alphabet A is 2.
Frequency of alphabet S is 3.
Frequency of alphabet I is 1.
Frequency of alphabet T is 2.
Frequency of alphabet N is 1.
 So the total number of letters in latter ASSISTANT is 9.
Now second later is STATISTICS so the frequency of each alphabet that is given below
Frequency of alphabet A is 1.
Frequency of alphabet S is 3.
Frequency of alphabet T is 3.
Frequency of alphabet I is 2.
Frequency of alphabet c is 1.
So the total number of alphabet in latter STATISTICS is 10.

Now we have to write the common alphabet in both letters
The common letters are S,A,T,I.
Now probability of each alphabet is found by dividing frequency of alphabet to total number of alphabet in that word.
So I make table for probability distribution here:

alphabet	ASSISTANT	STATISTICS
S	3/9	3/10
A	2/9	1/10
T	2/9	3/10
I	1/9	2/10

Probability for common latter is $(P)$ = $P(S) + P(A) + P(T) + P(I)$
\[\]$(P)$$ = \dfrac{3}{9} \times \dfrac{3}{{10}} + \dfrac{2}{9} \times \dfrac{1}{{10}} + \dfrac{2}{9} \times \dfrac{3}{{10}} + \dfrac{1}{9} \times \dfrac{2}{{10}}$
Now multiply numerator and denominator term we get
$P = \dfrac{9}{{90}} + \dfrac{2}{{90}} + \dfrac{6}{{90}} + \dfrac{2}{{90}}$
In this denominator is same so LCM is 90
$P = \dfrac{{9 + 2 + 6 + 2}}{{90}}$
$P = \dfrac{{19}}{{90}}$
So the probability of getting same number is $\dfrac{{19}}{{90}}$

Note: Remember one thing in this question we have to only take probability of common alphabet not all.