Question

A lottery ticket has a 5x5 grid of 25 scratch-off areas. Each area independently has probability 0.6 of revealing a star underneath. If there is a line of 5 stars in a row (either vertically or horizontally) when all the areas are scratched off, that line is a winner.

Let X_i = 1 if line i is a winner, and 0 otherwise,
where i = 1, 2, 3, 4, 5 are the vertical lines and i = 6, 7, 8, 9, 10 are the horizontal lines

a) (2) Find the expected value and variance of X_i.

b) (2) Find the expected number of winning lines on the ticket.

c) (2) Find the variance of the number of winning lines on the ticket.

d) (1) How would the results in (b) and (c) change if 5 stars in a diagonal line was also considered a winning line? (i.e. how many additional variables and non-zero covariances would there be?) No calculations are necessary.

Answer 1