Activity 5a. Tetris

Read the statistical experiment below and answer the question that follows.

Recall that Tetris pieces have seven possible shapes, labelled I, J, L, O, S, T, Z. Assume that, at any stage of the game, all seven shapes are equally likely to be produced, independent of whatever pieces have been produced previously.

Consider the sequences of three consecutive pieces observed during a game of Tetris, such as JLL, ZOS, ZSZ, III and so on.

a What is the probability that a sequence of three pieces does not contain a Z?

b Hence, what is the probability that a sequence of three pieces has at least one Z?

c What is the probability that a Z occurs in the first three-piece sequence observed?

d What is the probability that a Z does not occur in the first three-piece sequence, but does occur in the second such sequence? (Here we are considering non-overlapping sequences of three pieces.)

e. Hence, what is the distribution of X, the number of three-piece sequences observed before a three-piece sequence with a Z in it first appears? Write down its probability function PX (x).

f. Evaluate px (x) for x = 2,3,4,10.