Question

involving the flipping of a weighted coin. Assume that the coin is weighted so that Pr[H]=1/4, and that the coin is flipped until a head appears, or 3 consecutive tails appear. The random variable X is defined to be the total number of flips of the coin.

How many different values are possible for the random variable X?

And there is a 2nd part to this question only available after answering the 1st.

Answer 1

It is given that, the coin is flipped until a head appears, or 3 consecutive tails appear

Now, The random variable X is defined to be the total number of flips of the coin.

And we need to find how many different values are possible for the random variable X.

The, random variable X will assume values until either a head or three consecutive tails appear. Lets say H = Head and T = Tails

Thus, possible cases are

First head and thus we stop => H => 1 flip

First tails second head and thus we stop as we got head => TH => 2 flips

First & second tails and third head and thus we stop as we got head => TTH => 3 flips

First three are tails and we stop as we got three consecutive tails => TTT => 3 flips

Thus, there are 3 different values possible for the random variable X i.e.

{1,2,3}

As both 3rd case and 4th case requires 3 flips

Part b)

Total 4 possible scenarios

X =1, occurs 1 time (H). Thus, P(X =1) = 1/4

X =2, occurs 1 time (TH). Thus, P(X =2) = 1/4

X = 3, occurs 2 times (TTH, TTT). Thus, P(X =3) = 2/4 = 1/2

Thus, the probability function is

X	1	2	3
P(X)	1/4	1/4	1/2