Q-1) Two players A and B want respectively m and n points of winning a set of games ; their chances of winning a single game are p and q respectively, where the sum of p and q is unity ; the stake is to belong to the player who first makes up his set : determine the probabilities in favour of each player.
Q-2) In a purse are 10 coins, all shillings except one which is a sovereign ; in another are ten coins all shillings. Nine coins are taken from the former purse and put into the latter, and then nine coins are taken from the latter and put into the former. Find the probability that the sovereign is still in the first purse.
Q-3) A coin whose faces are marked 3 and 5 is tossed 4 times, what is the probability that the sum of the numbers thrown is less than 15?
Q-4) A bag contains a coin of value M, and a number of other coins whose aggregate value is m. A person draws one at a time till he draws the coin M, find the expected value that he can draw from the bag.
Q-5) A die is thrown three times, and the sum of the three numbers thrown is 15, what is the probability that the first throw was a four?
Q-6) A 7-digit number has 59 as the sum of its digits. Show that the probability of it being divisible by 11 is \(\frac{4}{21}\).
Q-7) On a straight line of length \(a\) two points are taken at random. For a given \(b\), what is the probability that the distance between them is greater than \(b\)?
Q-8) Suppose that a town has population of n people. A person spreads a rumour to a second, who in turn repeats it to a third and so on. Assuming that at each stage, the recipient of the rumour is chosen at random from the remaining (n-1) people, find the probability that the rumour will be repeated n times :
a) without it being repeated to any person
b) without it being repeated to the originator.
Q-9) A bag contains \(\frac{n(n+1)}{2}\) coins, of which one is marked 1, two are marked 4, three are marked 9, and so on. A person puts in his hand and draws out a coin at random, and is to receive as many rupees as the number marked up on the coin. What is the expected amount of rupees that the person can receive from this experiment?
Q-10) A player tosses a coin, and is to score 1 point for every head that turned up, and 2 points for every tail. He keeps doing this until his score reaches or passes n. Let \(p_n\) denote the probability that he attains exactly \(n\).
Show that \(2p_n=p_{n-1}+p_{n-2}\). Hence or otherwise, calculate \(p_n\).