Question

1. Suppose Jane has a fair 4-sided die, and Dick has a fair 6-sided die. Each day,
they roll their dice (independently) until someone rolls a “1”. (Then the person
who did not roll a “1” does the dishes.) Find the probability that …
a) they roll the first “1” at the same time (after equal number of attempts);
b) it takes Dick twice as many attempts as it does Jane to roll the first “1”;
c) Dick rolls the first “1” before Jane does.

Answer 1

1a) both dick and jane rolls 1 at the same time
1/4 x 1/6 = 1/24

b) if suppose jane rolls 1 on first attempt
1/4 x 5/6 x 1/6 = 5/144
if jane rolls 1 on 2nd attempt
3/4 x 1/4 x 5/6 x 5/6 x5/6 x 1/6 = 375/20736
∴the chance is 1/24 x (3/4)^n-1(5/6)^2n-1where n is the no. of attempts. I might be wrong on this

c) dick rolls 1 and jane does not
1/6 x 3/4 = 1/8

Answer 2