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You are correct; convenience sampling is not random sampling.
Simple Random Sample Stratified Random Sampling Cluster Sampling Systematic Sampling Convenience Sampling
Circular systematic sampling is a random sampling method. An example is random sampling of households. Assume that a random number generator provides the number 49 as a starting point. Starting with the household that is 49 on the target list, every nth household on the list would be sampled until the desired sample size is reached
stratified random sampling is a sample(strata) that a same and hemogenieous in group and that a different and heterogenious in group
They think that if they used the random sampling after people voted because they want to get a better view of who will be the next president.