Solution:

This puzzle is also known as Monty Hall Puzzle. If Michael choose the box on the table, the probability of finding the pearl inside will be 2/3.

Let us consider the same thing with more boxes. If Sarah placed 100 boxes on the table with one box that contains the pearl and Michael is allowed to pick random, 99 boxes will remain on the table. Now at least 98 boxes in the table will be empty. Sarah will remove these 98 boxes and now Michael will have to choose from the box on table or the box in his hand. If he chooses the box below, the probability of finding the pearl in the box will be 99/100.

Now you must be convinced why he should pick the box on the table.

Solution:

This is a bit tricky question that almost every time leaves the interviewee baffled.

Now take into consideration that the first friend can have his birthday in any of the months (there are 12 months all together).

All you have to find out is the probability that the second one will have his birthday in the same month which is a piece of cake.

The probability = 1/12.

Solution:

The worst case will be when you don't take out three pairs of same color will be that you take out 2 red, 4 yellow and 5 of each color. In that manner, you will have to take out 46 socks. But if you take out one more sock, you will have 6 socks of at least one color. Thus in that case, you must take out 47 socks to make sure that you have taken out at least three pairs of same color socks.

Solution:

If you have come up with the fact that the bugs can avoid the collision only when they decide to move in the same direction be it clockwise or anti clockwise, you have already solved half of it. In any other way, the bugs will definitely run into each other.

Now the bugs can move either clockwise or anti-clockwise. The probability of choosing a specific direction is then 1/2.

We will utilize the simple equation of probability to find out our answer

P(No Collision) = P (The bugs move in clockwise direction) + P(The bugs move in anti-clockwise direction)
= 0.5 * 0.5 * 0.5 + 0.5 * 0.5 * 0.5
= 0.25

Thus the probability that the bugs won't collide into each other is 0.25 or 1/4.

Solution:

One thing you must take note of is that in the question they are talking about pair of socks not individual socks.

Thus, total socks = (9x2) + (8x2) = 18 + 16 = 34
Total possibilities = 34*33*32/3*2*1 = 5984

The next part can be a little tricky. Read the question again. You are asked to calculate the probability that she will pick up a pair of blue socks. That also includes picking up three blue socks.

Thus our favorable events = select 2 blue + all 3 blue
= (18*17/2*1)(16/1)+ (18*17*16)/(3*2*1)
= 2448 + 816
= 3264

Hence the Probability = 3264/5984 = 6/11 = 0.54545

Solution:

This is rather a confusing logical puzzle which can leave you without an answer. The answer logically is quite simple though. It is entirely sure that Mark will definitely be at a certain point exactly at the same time on both of his trips.

To simplify and prove it we can send a hypothetical brother of Mark to their mother's house at the same time when Mark starts his journey back to their home. There will be a certain place where they will meet (as they are following the same route) and also they will be at that place the same time.

Thus no fact will matter in any case and there will be one point where Mark will be there at the same place at the same time in both of his journeys.

Solution:

This is a bit tricky series problem. Let us break that down for you

23 is the first term
25 = 21*1 + 2
53 = 25*2 + 3
163 = 53*3 + 4
657 = 163*4 + 5
3291 = 657*5 + 6

Thus the next term will be 3291 * 6 + 7 = 19753

Solution:

Let us simplify the question and then it will lead us to the answer easily. There are three possibilities. Thus the probability of choosing any door is 1/3rd. 1/3 of the time the door to heaven will be chosen by Amanda and thus 1/3 of the time it will take her zero days. 1/3 of the time, the day which leads to hell for one day will be chosen and of those the right door will be chosen the next day. So 1/9 trips take one day. In the similar manner, 1/9 will take to the door which takes to 2 days hell.

When that happens, the cases will be split again and again. Thus on average, it will take three days for Amanda to reach Heaven by choosing the right gate.

Solution:

Consider the fact that both Andy and Brian can make independent decisions. Thus we will have to apply Intersection here. First let us calculate their individual probability.

Probability of Andy speaking truth = 80/100 = 0.8
Probability of Brian speaking truth = 60/100 = 0.6

Probability of Andy Intersection Probability of Brian = 0.8 * 0.6 = 0.48

Solution:

If you pick '2 2 2' or any combination of '1 2 3' then the sum will add up to 6.

Now the probability that you will get '2 2 2' is 5/16 (5 2s out of 16) times 4/15 (4 2s left out of 15) times 3/14 (3 2s left out of 14).
Thus the total probability is 5/16 * 4/15 * 3/14 = 1/56.

The probability of getting any combination of '1 2 3' is equal to 6/16 (6 1s in the list) times 5/15 (5 2s out of 15 numbers left) times 5/14 (5 3s out of 14 numbers left) times the number of different combinations of '1 2 3' which is 6.
Thus the total probability is 6/16 * 5/15 * 5/14 * 6 = 15/56.

1/56 + 15/56 = 16/56 or 2/7 or 28.6%.

Thus the probability of choosing the numbers which sums up to 6 is 29 percent (approx.)