n books are to be arranged on a self. These include m volumes of a science book (m < n). The probability that in any arrangement, the volumes of science books are in ascending order is?

From the set of odd number < 100, if we exclude multiples of 5, we get the set of numbers < 100 and relatively prime to 100.

The number of such numbers = 50 - 10 = 40
? Reqd. Probability = 40 / 100 = 2/5

Corresponding to n tosses, the Probability of getting no head = (1/2)ⁿ and,
Therefore, the probability of getting at least one head = 1 - (1/2)ⁿ
Now, 1 - (1/2)ⁿ ? 99/100
? (1/2)ⁿ ? 1/100
? n ? 7

? p = 1/5
? q = 1 - 1/5 = 4/5
The probability that none will hit in 10 shots = (4/5)¹⁰
? Reqd. probability = 1 - (4/5)¹⁰

? Probability in each trial (shooting) = 0.3
? Reqd. probability = (0.3)¹⁰

P(M) = 1/4; P(W)= 1/3
? P (M) = 1 - 1/4 = 3/4
? P(W)= 1 - 1/3 = 2/3

Reqd. probability = P(M) P(W) = 3/4 x 2/3 = 1/2

? Probability that no man out of n men aged x years will die in a year = (1 - p)ⁿ
? Probability that out of n men at least one will die in a year = 1 - (1 - p)ⁿ .
When at least one, man dies, any one out of the n men may be the first to die

? Probability of machine failing during a day p = 0.95
? q = Probability of its working during a day = 1 - p = 1 - 0.95 = 0.05

Required probability = q⁴ = (0.05)⁴
= 0.00000625

n(S) = ¹⁰C₇ = ¹⁰C₃
n(E) = ⁸C₃
? P(E) = n(E)/n(S) = 7/15

Probability of multiple of 2 = 3/6 = 1/2
Probability of multiple of 3 = 2/6 = 1/3
Since there are two dice.
? The required probability = 2 x 1/2 x 1/3 = 1/3

Total no . of cases = ⁶P₂ = 6 x 5 = 30
Non-favourable cases are (4, 5) , (5, 4), (4, 6), (6, 4), (5, 6), (6, 5)
? Probability that event will not happen = 6 /(6 x 5) = 1/5
? Reqd. probability = 1 - 1/5 = 4/5