8/11/2019 Further maths distribution.pdf

1/2

Two-Player Geometric Distribution Past Year Questions

1. A fair die is thrown until a 5 or a 6 is obtained. The number of throws taken is denoted by therandom variable X. State the mean value of X.Find the probability that obtaining a 5 or a 6 takes more than 8 throws.

Find the least integer n su h that the probability of obtaining a 5 or a 6 in fewer than n throws ismore than !."".

#. A hess player$ X$ plays a series of games against an opponent %. For ea h game$ the probabilitythat X wins is p$ independently of the results of other games. &rite down the probability$ in termsof p and n $ that X's first win is in the n th game.

The number of games up to and in luding X's first win is denoted by N.

(a) &rite down *(+) and ,ar(+) in terms of p$ and dedu e *(+ #).

(b) Find -(1 + a )$ where a is a positive integer.( ) /f X plays 0 games against %$ show that the probability that this series of games ontains at least #

onse utive wins by X is p #( 2#p).

. A bulb is tested by swit hing on and off repeatedly. *a h time the bulb is swit hed on$ theprobability that it fails is 13#!!. Assuming independen e$ find the probability that the bulb is stillburning after it has been swit hed on 1!! times.

The bulb fails when it is swit hed on for the Nth time. &rite down the value of *(+)$ and find thegreatest value of n su h that -(+4n) is less than !.1.

0. n average 1 in 8 people in a parti ular ommunity is left2handed and the rest are right2handed. Asample of people is hosen at random$ one by one$ until a left2handed person is obtained. Findthe probability that the number of people in the sample is less than 1#.

Another sample is hosen at random$ one by one$ until at least one left2handed and at least one2right handed person have been obtained. The number of people in this sample is denoted by +$ sothat the sample onsists of

*ither (+21) left2handed people followed by one right2handed personr (+21) right2handed people followed by one left2handed person

btain -(+7r)$ for r #$ and show that .5. An author sends his first manus ript to a large number of publishers$ 9$ :$ *$ ;$ in turn$ only

approa hing ea h one$ after the first$ if the one before has refused it. There is a onstantprobability < that his manus ript will be a epted by ea h publisher approa hed. The randomvariable = is the number of publishers approa hed$ up to and in luding the one who a epts themanus ript. &rite down the value of *(=) and ,ar(=)$ an al ulate

8/11/2019 Further maths distribution.pdf

2/2

For his se ond manus ript$ the author de ides to approa h two other publishers$ A then >$ forea h of whom the probability of a eptan e is ?$ before he approa hes 9$ :$ *$; . The probabilityof a eptan e by ea h of these publishers remains $ 9$ :$ *$ ; approa hed$ up to and in luding the one who a epts this manus ript. &rite downthe first few terms of the series for *(=) and *(+). >y omparing orresponding terms$ after the

se ond$ and using your value for *(=)$ show that

@se a similar method to show that $ and hen e find ,ar(+).6. Ale and >ill take alternate turns at ki king a football at a goal$ and their probabilities of s oring a

goal on ea h ki k are p1 and p2 respe tively$ independently of previous out omes. The fire personto s ore allows the other person one more ki k. /f the other then s ores$ the game is drawn. /f theother then misses$ the first has won the game. Ale begins a game. Show that the probability that

(a) Ale s ores first$ on his nth ki k$ is (b) Ale wins the game is

( ) The game is drawn

iven that $ find the eBpe ted total number of ki ks in the game.C. An ar her shoots at a target. /t may be assumed that ea h shot is independent of all other shots

and that$ on average$ she hits the bull's2eye with shots in #!. Find the probability that shereDuires at least 6 shots to hit the bull's2eye.&hen she hits the bull's2eye for the third time$ her total number of shots is %. Show that

! "!#

Simplify $%$ % and hen e find the set of values of r for whi h .:edu e the most probable value of %.