Expected Value of Infimum of Bernoulli

Question

Let $(\Omega,\mathcal{F},\mathcal{P})$ a probability space, consider a succesion of Bernoulli i.i.d. r.v with parameter $\theta=0.52$ I want to compute the expected value of $T(\omega)$ , where $T(\omega)$={$inf$ $0\leq n$, $X_n=1$}

I known by elemental probability that the succesion of Bernoullis is a Binomial(n,p) but the part of the infimum confuse me a bit. Anny suggestion?

Answer 1

Hint: $$\mathbb{E}(\tau) = \sum_{n=0}^{\infty} k \mathbb{P}(\tau = k)$$

And for $\{\tau = k\} = \{X_1 = 0, \ldots,X_{k-1} = 0, X_k = 1\}$