Questions

1. Three distinct numbers are selected from 100 natural numbers. What is the probability that all the three numbers are divisible by 2 and 3?
2. There are four letters and four addressed envelopes. What is the chance that all letters are not dispatched in the right envelope?
3. If five cards are selected at random from a standard 52 card deck, what is the probability of getting a full house.
4. A divisor of 2700 is selected. What is the probability that it is a multiple of 4 but not 5?
5. Robot is 3 cm from the left of the table and 17cm from the right of the table. The robot moves with equal probability 10cm to the left or to the right. What is the average number of steps for the robot to fall of off the table?
6. You and your friend are playing a game with a fair coin. The two of you will continue to toss the coin until the sequence HH or TH shows up. If HH shows up first, you win, and if TH shows up first your friend win. What is the probability of you winning the game?
7. The letters of the word 'ASSASSIN' are written down at random in a row. What is the probability that no two 'S' occur together?
8. Out of 21 tickets market with numbers from 1 to 21, three are drawn at random. What is the probability that the numbers form an Arithmetic Progression?
9. Three distinct numbers are selected from first 30 natural numbers. What is the probability that their sum is divisible by 3?
10. Twelve coins are divided into four groups of three coins each. The coins in each group have probabilities of landing heads equal to $p=\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{5},\dfrac{1}{9}$, respectively. All coins are tossed independently. Find the probability that the total number of heads obtained is odd.
11. How many distinct paths are there from the grid point (0,0) to the grid point (6,6) on a 6×6 lattice if you can only move to adjacent grid points by going one unit right or one unit up?
12. On average, how many cards in a normal deck of 52 playing cards do you need to flip over to observe your first ace?
13. You are bidding for a painting whose true value is uniformly distributed between $0 and $100,000. You submit a single bid b. If your bid exceeds the true value of the painting, you win the auction, pay b, and immediately resell the painting to an art museum for 1.5 times its true value. What bid b maximizes your expected profit? If no positive bid yields a profit, submit a bid of $0.
14. You roll a fair 6-sided die twice. Calculate the probability that the value of the first roll is strictly less than the value of the second roll.
15. Let $x_0 = 0$ and let $x_1, \dots, x_{10}$ satisfy$|x_i - x_{i-1}| = 1 \quad \text{for } 1 \le i \le 10,$ with $x_{10} = 4$. How many such sequences are there satisfying these conditions?
16. You toss a fair coin 100 times and record the outcomes. How many runs will you have on average? A run is classified as the longest continuous flips of heads or tails.
17. You are given 9 identical-looking coins, one of which is heavier than the other eight, and using a balance scale no more than twice, determine which coin is the heavier one.
18. There are 25 horses among which you need to find out the fastest 3 horses. You can conduct a race among at most 5 to find out their relative speed. At no point can you find out the actual speed of the horse in a race. Find out the minimum no. of races which are required to get the top 3 horses
19. You are given four identical bottles, exactly one of which contains a deadly poison. The poison will kill a rat exactly one hour after it drinks from the poisoned bottle; the other bottles are harmless. You may give drops from any bottles to any number of rats at time $t=0$. You may observe the rats only once, exactly one hour later. What is the minimum number of rats required to determine with certainty which bottle contains the poison? Explain your reasoning.