Likelihood of a single number being even: 1/2
Likelihood of four numbers all being even: 1/16
Likelihood of four numbers containing at least one odd number: 15/16
Number of consecutive four-number sequences in 1000 numbers: 997
Likelihood of all of these sequences containing at least one odd number: no idea, actually. Does the fact that every odd number interrupts four sequences invalidate the naive calculation?
Naive calculation: (15/16)^997 = 1.13595877459903082666e-28 = 0.000000000000000000000000000113595877459903082666