Suppose I have five biased coins $c1, c2, c3, c4$ and $c5$ each with different probabilities of getting heads.
- Pr. heads for coin $c1=0.5$
- Pr. heads for coin $c2=0.8$
- Pr. heads for coin $c3=0.6$
- Pr. heads for coin $c4=0.4$
- Pr. heads for coin $c5=0.2$
Each coin is flipped once in numerical order, and it takes me $10$ seconds to flip each coin. When you get a head you stop and that cycle is over.
Let $T$ be a random variable that describes the time of the event that you get a head; hence, $T$ can take on values ${10, 20, 30, 40, 50}$. I am interested in modeling the problem with survival analysis and I want to calculate the hazard function for each outcome.
The discrete hazard function is defined as follows
$$ \lambda_j=Pr(T=t_j|T \geq t_j) = \frac{f_j}{S_j} $$
To illustrate lets calculate the hazard for $t=30$ sec - which means that I have to get a head only on the third coin $c3$
$$ \lambda_j=Pr(T=t_c3|T \geq t_c3) = \frac{Pr(C1=tails)Pr(C2=tails)Pr(C3=heads)}{Pr(C3=tails)Pr(C4=tails)Pr(C5=heads)+Pr(C3=tails)Pr(C4=heads)+Pr(C3=heads)} $$
At first I was confused since $p$s were different and I could not use a distribution i.e.: Geometric. So is this the correct approach to this problem I created? Also, is there another distribution or shortcut that I can use? Something vague about renewal processes comes to mind.
Thank you