Am assuming it will be wrong to say probabilities that the person aged x dies in 10 and 20 years will be the same. Contingencies .

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You are correct, saying \(q_{x}\) is the same for all ages is saying that the probability of dying within the next year, regardless of your age, is the same.

Now, if \(l_0=1 000 000\) then to find out how many people are alive at age \(30\) we need to multiply \(l_0\) by the probability of surviving \(30\) years. Since \(q_x\) is the same for all ages, \(1-q_x=p_x\) is also the same for all ages.

Consider finding \(l_1\): \(l_1=l_0 \times p_0\).

Applying this to \(l_{30}\), we get that \(l_{30}=l_0 \times p_0 \times p_1 \times ... \times p_{29}=l_0 \times \left(p_x\right)^{30}=l_0 \times \left(1-q_x\right)^{30}\)