$_{k|}q_x = 0.02(k+1)$ for $k = 0, 1, 2, 3$ and $4$, where $_{k|}q_x$ means that a person of age $x$ will survive $k$ years and dies within $1$ year.
So I have that: $_4P_x= (P_x)(P_{x+1})(P_{x+2})(P_{x+3})$
and that $_{k|}q_x=(_kP_x)(q_{x+k})$
But I'm honestly just lost on how to convert $_{k|}q_x$ into $P$ (probability of the person surviving).
Any help greatly appreciated
The idea is to observe that ${}_4 p_x$ is the probability that a life aged $x$ survives at least $4$ years, so that $1 - {}_4 p_x$ is the probability of not surviving to $(x+4)$. But this is simply $$1 - {}_4 p_x = {}_{0|}q_x + {}_{1|}q_x + {}_{2|}q_x + {}_{3|}q_x,$$ since if $(x)$ does not survive to $(x+4)$, he dies at some time $k + t$ for $k = 0, 1, 2, 3$ and $0 \le t < 1$. These are precisely the failure probabilities given. Note that you should not include ${}_{4|}q_x$ because this is a failure probability for a life aged $x$ surviving $4$ years, then dying in the following year, which is not what we want--$(x)$ must die before reaching $(x+4)$.
In general, we can see that the relevant formula to apply in this case is $${}_n q_x = \sum_{k=0}^{n-1} {}_{k|}q_x.$$ Then we use $${}_n q_x + {}_n p_x = 1,$$ and the rest is straightforward.