If you still stuck in part a), you may think of the following way first:

Let [math] Y_i \sim \text{Bernoulli}(0.3) [/math] be the indicator of the [math] i [/math]-th toss having a head such that [math] X = \sum_{i=1}^{10} Y_i [/math].

Can you express the outcome after the 1st toss? If you are not sure, write it down with something like

[math] \begin{cases} W_H \text{ if } ~~ Y_1 = 1 \\ W_T \text{ if } ~~ Y_1 = 0 \end{cases} [/math]

in terms of the initial wealth [math] W [/math]. (assume the question have not given the number 1024 yet).

Note that while [math] Y_i [/math] is the indicator for head, we can also use [math] 1 - Y_i [/math] as the indicator for tail. Two common way to combine the above written cases

1. [math] W_HY_i + W_T(1 - Y_i) [/math] in an additive model

2. [math] W_H^{Y_i}W_T^{1-Y_i} [/math] in a multiplicative model

You should be familiar with the second one if you have learned something about the Binomial.