Tossing a coin until it lands head up

[HappyN]

A coin with unknown probability p of landing head up when tossed, was tossed until it first landed head up. It happened on the 5th toss. Describe an appropriate statistical model for this experiment and write down the observed Likelihood function L(p).

I assumed this was the same as modelling P(X1=...=X=4=0, X5=1) = (1-p)^4 (p)
Where 0 = tails and 1= heads
So the likelihood function would be L(p) = (1-p)^4(p)
But I don't think this takes into consideration the fact that the coin is being tossed just until it lands head up - any input would be greatly appreciated :)

 

[D H]

But I don't think this takes into consideration the fact that the coin is being tossed just until it lands head up - any input would be greatly appreciated :)

It does. Suppose instead the experiment was to toss the coin 5 times and report the number of times the coin showed heads. The event #heads = 1 for this experiment has a different probability than does the event #tosses = 5 for the experiment "toss the coin until heads appears and report the number of tosses."

The former experiment can be generalized to tossing the coin a fixed number n times, with the outcome of the experiment characterized by the number of heads r that appeared during the run of the experiment. The probability distribution for this experiment is given by the binomial distribution. The latter experiment can be generalized to tossing the coin repeatedly until r heads have appeared. The outcome of the experiment characterized by the number of tosses n needed to make those r heads appear. The probability distribution for this experiment is given by the negative binomial distribution.

You correctly wrote the negative binomial distribution specialized to the case r=1 and n=5. In other words, you implicitly did "take into consideration the fact that the coin is being tossed just until it lands head up."