A fair die and a fair coin are being tossed simultaneously a

A fair die and a fair coin are being tossed simultaneously and repeatedly.

(a) What is the probability that in the first k tosses we won

Solution

1) probability of getting a head or tail is 1/2....so in the first k tosses..the probabilty of getting no tail will be (1/2)^k.

probabilty of getting odd or even number is 3/6=1/2......so in the first k tosees...the porbabilty of getting even outcome will be (1/2)^k....so answer will be multiplication of both as both are independent events=(1/2)^2k.

2)uptill k-1 th toss....head comes=(1/2)^k-1.Now at k^th toss, tails comes..so (1/2)*(1/2)^k-1 =(1/2)^{k .}Similarly for die= (1/2)^k. Now as \'or\' is mentioned...any event occurs is fine...So we add them i.e (1/2)^k *2=(1/2)^k-1.

3)

f(x)=(1/2)^x-1 for x>1 and f(x)=1/2 for x=1 and f(x)=0 for x=0 ( Here x=1 and x=0 are conidered special cases where direct (1/2)^x-1 doesn\'t give the right answer)