Geometric Distribution Coin Flip

[dspampi]

Consider the following experiment: a coin that lands heads with
probability p is flipped once; if on this first flip it came up H, it is then repeatedly flipped until a T occurs; else, if on the first flip it came up T, it is then repeatedly flipped until a H occurs. Let X be the total number of flips. Find the p.m.f. of X, and the mean of X.


Ok so I know this has to be a geometric distribution because we will repeat the experiment until we get a different side of the coin.

Probability h = p and t = (1-p);
Does this mean you have to condition for the two different cases and sum them their probabilites?

Aka Px(k|1st T) = p* (1-p)^(1-k)

and Px(k|1st H) = (1-p)*(p)^(1-k)?

 

[Ray Vickson]

Consider the following experiment: a coin that lands heads with
probability p is flipped once; if on this first flip it came up H, it is then repeatedly flipped until a T occurs; else, if on the first flip it came up T, it is then repeatedly flipped until a H occurs. Let X be the total number of flips. Find the p.m.f. of X, and the mean of X.


Ok so I know this has to be a geometric distribution because we will repeat the experiment until we get a different side of the coin.

Probability h = p and t = (1-p);
Does this mean you have to condition for the two different cases and sum them their probabilites?

Aka Px(k|1st T) = p* (1-p)^(1-k)

and Px(k|1st H) = (1-p)*(p)^(1-k)?

Yes, it means that; however, you need (1-p)^(k-1) and p^(k-1), not what you wrote.

RGV