Solution:

0.1667 or 16.67%

When rolling two fair six-sided dice, there are 36 possible outcomes (6 outcomes for the first die multiplied by 6 outcomes for the second die).

To calculate the probability of rolling a sum of 7, we need to determine the number of combinations that result in a sum of 7. These combinations are:

1 + 6
2 + 5
3 + 4
4 + 3
5 + 2
6 + 1

So, there are 6 favorable outcomes that result in a sum of 7.

The probability of rolling a sum of 7 is calculated by dividing the number of favorable outcomes by the total possible outcomes:

Probability = (Number of Favorable Outcomes) / (Total Possible Outcomes) = 6 / 36 = 1/6 ≈ 0.1667

Therefore, the probability of rolling a sum of 7 with two fair six-sided dice is approximately 0.1667 or 16.67%.

Solution:

In Mansion M2, we know that older child is boy, therefore there are two cases
Older Younger
Boy Girl
Boy Boy
Probability of at least one girl child: 1/2 => 50%

In Mansion M1 there are three cases
Older Younger
Boy Girl
Boy Boy
Girl Boy
Probability of at least one girl child: 2/3 => 66.67%

Solution:

4%

Explanation:
The winning chance probability is still 4% as the outcome of Samuel winning the lottery is a separate event from the 1st four losing it.

Solution:

1/3

Explanation
Assuming T1 is the Toy for brother1, T2 is the toy for brother2 and T3 is the toy for brother3.

Following are the possible cases for toys distribution:

Boy1 Boy2 Boy3
T1 T2 T3
T1 T3 T2
T2 T1 T3
T2 T3 T1 .... (A)
T3 T1 T2 .... (B)
T3 T2 T1

In both steps (A) & (B), no one gets the correct toy.
Therefore probability that none brother can get the own toy is 2/6 = 1/3

Solution:

1/6227020800

Explanation
This can be solved through a simple mathematical equation.

N cards have n! permutations.
Therefore, the probability of obtaining a specific one by chance is 1/n!

Now, in this particular case, n = 13
Therefore, the required probability is 1/6227020800

Solution:

The probability is 5/36.

Event = {(1,5) , (2,4), (3,3), (4,2), (5,1)}
Therefore, P (getting sum = 6) = 5/36

Solution:

36/71.

Explanation:

Solution:

India has the highest chances of winning the war (47/90)
Missile Firing order:
Chances of Remaining Unhit - Pakistan:
Chances of Remaining Unhit - Srilanka:
Chances of Remaining Unhit - India

Pakistan - Srilanka - India
1/2 0 1/2

Pakistan - India - Srilanka
1/2 0 1/2

India - Pakistan - Srilanka
1/10 16/45 49/90

Srilanka- Pakistan - India
1/2 0 1/2

Srilanka - India - Pakistan
1/10 16/45 49/90

India - Srilanka - Pakistan
1/10 16/45 49/90

Total chances of being unhit (sum of the probabilities divided by 6):
27/90 16/90 47/90

Solution:

There is a chance of 1/3 to be its best in all the shots. The reason for this is that the shots do not depend on the shot that is made earlier. The chance for the third shot to be its best is 1 minus the chance that the last shot is the best shot. Therefore the answer for this question is 2/3.
There are also other possibilities. There are three arrows A, B, and C. the B arrow is away from the arrow A. so after the shot A the following will be the outcome or the possibility with the shots.
Best shots: A, A, C
Worst shots: C, B, B
From all the three outcomes, the arrow C is worse than the arrow A.

Solution:

The best way to increase the probability of winning is to place one yellow ball in the first bowl and all the other balls in the second bowl.

Now, you have a 1/2 probability of choosing the bowl with just one yellow ball. If you choose that, you win.

If you choose the other bowl, it has 49 yellow balls and 50 white balls. You still have almost 1/2 probability of picking a yellow ball out of it and winning.