Question

What is the probability that the sum of any two different single digit natural numbers is a prime number?

• A. $\frac{5}{27}$
• B. $\frac{7}{18}$
• C. $\frac{1}{3}$
• D. None of the above

Answer 1

Answer: (B)

$\frac{7}{18}$

set of single digit natural numbers includes $N:1,2,3,4,5,6,7,8,9$

...

now, the maximum sum that you can get is $9+9=18$ ( since it's not mentioned in the qurstion , so we can use same digit twice ).

So,

primes numbers that we can have are,$2,3,5,7,11,13,17$

Now solve for different prime numbers!

...

$1+1$

$1+2$

$1+4$

$2+3$

$1+6$

$2+5$

$3+4$

$2+9$

$3+8$

$4+7$

$5+6$

$4+9$

$5+8$

$6+7$

$8+9$

[ Since there is no condition on picking up the numbers, hence 1+2 and 2+1 is same ]

So,

These are the only 15 cases where we'll have a prime number...

Total number of ways of adding two single digit natural numbers are $=9+8+7+6+........+1=45$

...

Hence the probability is $\frac{15}{45}=\frac{1}{3}$

...

*** Now here is a thing, if you are looking for two different natural numbers,

Then;

The probability will be $\frac{14}{36}=\frac{7}{18}$