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I was asked to come up with a likelihood function based on the following problem (summarized):

A student wants to know how many people she will have to ask on average before she has a sample of 20 males and 20 females from the USA. She records the number of males she asked before finding a male from USA, and the same for females. She stops asking after a set time, whether or not the last person she asked was from the USA.

As the problem requires the number of trials before reaching a success, I thought geometric function was the right choice. She doesn't necessarily stop when she reaches 40 (male + female) successes, so this is the likelihood function I came up with:

$$ \prod_{i=1}^n(1-p(x_i;\beta))^{y_i-1}p(x_i;\beta) $$

where I parametrized the probability as:

$$ p(x_i;\beta)=\beta_0+\beta_1x_i $$

$x_i$ being 0 if male and 1 if female,

$y_i$ the number of people asked until ith observation,

$\beta=(\beta_o,\beta_1)$ is the unknown parameter vector.

However, the answer sheet has parametrized the probability function as:

$p(x_i;\beta)=\frac{1}{(1+exp^{-\beta_0-\beta_1x})}$

I do not understand why the probability function has been parametrized as the logistic function. Am I missing something here or is what I wrote above also a viable solution?