To solve this problem, we need to use the binomial distribution formula. The binomial distribution formula is given by:

P(X=k) = C(n, k) * p^k * (1-p)^(n-k)

Where:
- n is the number of trials (in this case, the number of times the dice is rolled: 180180)
- k is the number of successful outcomes (in this case, the number of times face 44 turns up: at least 3535)
- p is the probability of a successful outcome (in this case, the probability of face 44 turning up on a single roll)

Since the dice has six faces, each face has an equal probability of turning up, and there are two face 44 on the dice, the probability of face 44 turning up on a single roll is 2/6 = 1/3.

Using the binomial distribution formula, we can calculate the probability of face 44 turning up at least 3535 times:

P(X>=35) = P(X=35) + P(X=36) + ... + P(X=180180)

P(X>=35) = ∑(k=35 to 180180) C(180180, k) * (1/3)^k * (2/3)^(180180-k)

Calculating this sum would require a lot of calculations and would not be feasible manually. To solve this, we can use a normal approximation to the binomial distribution.

For large values of n and when both np and n(1-p) are greater than or equal to 5, the binomial distribution can be approximated by a normal distribution with mean μ = np and standard deviation σ = sqrt(np(1-p)).

In this case, np = 180180 * 1/3 = 60060 and n(1-p) = 180180 * 2/3 = 120120.

We want to find the probability of face 44 turning up at least 3535 times, so k >= 35.

To use the normal approximation, we can transform the inequality:

P(X>=35) = P(X>34) = 1 - P(X< />

Using the normal distribution, we calculate the z-score for k = 34:

z = (k - μ) / σ = (34 - 60060) / sqrt(60060 * (1/3) * (2/3))

We can then use a standard normal distribution table or calculator to find the cumulative probability P(Z<=z) and="" subtract="" it="" from="" 1="" to="" find="" the="" probability="" of="" p(x="">34).

Note: The normal approximation assumes that the conditions for approximation are met, so the result may not be exact but should be a good approximation given the large number of trials.

Unfortunately, without a specific value for k (the number of times face 44 turns up), we cannot provide an exact probability.