In a certain game of chance, a square board with area 1 is colored with sectors of either red or blue. a player, who cannot see the board, must specify a point on the board by giving an x-coordinate and a y-coordinate. the player wins the game if the specified point is in a blue sector. the game can be arranged with any number of red sectors, and the red sectors are designed so that calculate the minimum number of red sectors that makes the chance of a player winning less than 20%.

85.560%

step-by-step explanation:

conditional probability is defined as:

p(a|b) = p(a∩b) / p(b)

in other words, the probability that a occurs, given that b has occurred = the probability that both a and b occur / the probability that b occurs.

in this example:

p(turns 64 | turns 44) = p(turns 64 & turns 44) / p(turns 44).

if someone turns 64, they must have already turned 44, so:

p(turns 64 | turns 44) = p(turns 64) / p(turns 44)

on page 42, we see that of the original 100,000, 94944 turn 44, and 81234 turn 64.   therefore:

p(turns 64 | turns 44) = 81234 / 94944

p(turns 64 | turns 44) = 0.85560