This question asks you to investigate the number of throws of a die up to and including a certain result.
Throw an unbiased 6-sided die until you get the first 6. Let X= number of throws.
(a). (i) Find P(X=1).
(a). (ii) Explain why P(X=4)=56316.
(a). (iii) Find P(X<9).
Let p=56, q=16
(b). (i) Express E(X) as an infinite series with p and q.
E(X)=q+2pq+3p2q+4p3q+⋯
E(X)=q+pq+p2q+p3q+⋯
(b). (ii) By using the fact that q=1-p, or otherwise, find E(X).
(c). (i) Show thatVar(X)=2p1+2p+3p2+⋯+1+p+p2+p3+⋯-36
(c). (ii) By considering ddtt+t2+t3+⋯, or otherwise, find Var(X).
Throw the die until you get the same score two consecutive times. Let Y= number of throws.
(d). (i) Find P(Y=2).
(d). (ii) Find P(Y=5).
(d). (iii) Find E(Y).
Throw the die until you get each score at least once. Let Z= number of throws.
(e). (i) Find P(Z=6).
(e). (ii) Find P(Z=7).
(e). (iii) Find E(Z).