Suppose that the number of persons who visit a yoga studio arrive in accordance with a Poisson process having rate $\lambda$ (day as a unit). Suppose further that each person who visits is, independently, female with probability $p$ or male with probability $1-p$. Let $N_1(t)$ and $N_2(T)$ denote respectively the number of females and males who visit the yoga studio by the time $t$. Further $S_n^{1}$ and $S_n^{2}$ be respectively the arrival times of the nth female and male visitors.
I did part a and b of this question already, the results are
a) The probability that the first person who visits the yoga studio is male is $1-p$
b) Fix $n\geq 1$, the probability that the first n persons who visits the yoga studio are all female is $p^n$
c) Suppose that anyone who visits the yoga studio will stay for at least one day. What is the probability that at any time during the first day,there are always more women than men in the studio ?
d) Denote by $S_n$ the arrival time of the th visitor to the yoga studio. Let $$K = min\{k_1+k_2 : S_{k_1}^{1}<S_{k_2}^{2}<S_{k_1+1}^{1}<S_{k_2+1}^{2}<S_{k_1+2}^{1}\}$$ Compute $E[S_k]$.
I am currently stuck at parts c and d. For part c, I am thinking that the ballot problem has some sort of connection, but not sure how to use it. For part d, I don't even know what the question is trying to ask
Any help is appreciated, thank you