Probability

JEE Main 2025 (Online) 2nd April Morning ShiftNumericalHard📈 Topic in 37% papers+4

Ques. 1JEE Main 2025, 2 April (Morning Shift)

Three distinct numbers are selected randomly from the set \1, 2, 3, , 40\. If the probability, that the selected numbers are in an increasing G.P., is mn, (m, n) = 1, then m + n is equal to .

Answer

2477

Solution

Total ways = 40C3 = 9880

Let the G.P. be a, ar, ar2 with r = pq > 1, (p, q) = 1.

For all terms to be integers, a = kq2; terms are kq2, kpq, kp2 and we need kp2 40.

Number of such G.P.s for each p is (number of q < p coprime to p) [ 40p2 ]:

p = 2 (q = 1): 1 10 = 10

p = 3 (q = 1, 2): 2 4 = 8

p = 4 (q = 1, 3): 2 2 = 4

p = 5 (q = 1, 2, 3, 4): 4 1 = 4

p = 6 (q = 1, 5): 2 1 = 2

p 7 p2 > 40: none.

Favourable = 10 + 8 + 4 + 4 + 2 = 28

Probability = 289880 = 72470 (2470 = 2 5 13 19, so already in lowest terms)

m + n = 7 + 2470 = 2477

Answer: 2477

Keep practising — Probability

Ques. 2JEE Main 2025, 28 January (Morning Shift)

Two numbers k1 and k2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1 + ik2, (i = -1) is non-zero, equals

Ques. 3JEE Main 2025, 28 January (Evening Shift)

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

Ques. 4JEE Main 2025, 23 January (Morning Shift)

One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is

Ques. 5JEE Main 2025, 23 January (Evening Shift)

A board has 16 squares as shown in the figure : [figure] Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

35
45
2330
710

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