The probability that a man lives after 10 years is 1/4 and that his wife is alive after 10 years is 1/3. The probability that neither of them is alive after 10 years is?

? Probability for 3 = (1, 2), (2, 1) = 2/36
? Probability for 5 = (1, 4), (2, 3), (3, 2), (4, 1)= 4/ 36
? Probability for 11 (5, 6), (6, 5) = 2/36

? Reqd . Probability = 2/36 + 4/36 + 2/36 = 8/36 = 2/9

First we place seven white balls at places market W.
W -- W --W --W --W -- W -- W

If we place three black balls at -- places. Then no two black balls will be placed adjacently.
Total No, of -- places = 8

? No. of favourable ways for black balls = ⁸C₃ = 8 x 7 x 6 / 1 x 2 x 3 = 56
Total no. of equally likely cases = 10! / 7!3!
= (10 x 9 x 8) / (1 x 2 x 3) = 15 x 8 = 120

? Reqd. Probability = 56 / 126 = 7/15

There are 13 hearts and 3 more kings
? p (heart or a king ) = (13 + 3)/52 = 4/13
? P (neither a heart nor a king) = 1 - 4/13 = 9/13

Clearly, n (S) = 52, there are 4 kings and 4 queens
? P(E) = n (E) / n (S) = 8/52 = 2/13

Clearly n(S) = 52. There are 26 red cards (including 2 kings ) and there are 2 more kings.
Let (E) be the event of getting either a red card or a king .
Then, n(E) = 28
? P(E) = n(E) / n(S) = 28/52 = 7/13

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

? 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

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

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 each arrangement of (n - m) other books, there is a unique arrangement of the m volumes of the science book in ascending order and m! arrangement of the m volumes in random order
&Reqd. Prob = p = 1/m!