Q. In an examination mark obtained by Arun is between 10 to 24, and marks obtained by Varun is between 7 to 19. Marks obtained by Varun are a prime number while marks obtained by Arun are even integers.
Which of the following statement(s) is/are True?
I. Probability of sum of marks obtained by both is prime number is 1/2
II. Probability that difference of marks obtained by both is even number, is 0
III. Probability of sum of marks obtained by both, is perfect square is 1/6
A. II only
B. I and II only
C. II and III only
D. I only
E. None of these
Ans: I and II only
Sol:
Marks obtained by Arun = [12, 14, 16, 18, 20,
22]
Marks obtained by Varun = [11, 13, 17]
Possible sum of marks = [23, 25, 29, 25, 27, 31,
27, 29, 33, 29, 31, 35, 31, 33, 37, 33, 35, 39]
I. Probability of sum of marks obtained by both is
prime number is 1/2.
In 10 cases we get sum is prime = [23, 29, 31,
29, 29, 31, 31, 31, 37]
Total outcomes = 18
Required probability = 9/18 = 1/2
II. Probability that difference of marks obtained
by both is even number is 0
Marks obtained by Arun are an even number,
while Varun is an odd number.
Even – odd = odd
So required probability = 0
III. Probability of the sum of marks obtained by
both is a perfect square.
In two cases we get sum is perfect square = [25,
25]
Required probability = 2/18 = 1/9
Only I and II is true
Hence answer is option B