Can I play regular xbox 360 games like bioshock on an xbox 360 arcade, and be able to save them?

Can I play regular xbox 360 games like bioshock on an xbox 360 arcade, and be able to save them? Blog: BioShock Infinite revolutionises UE3 · If I buy xbox arcade games using my gamertag on my sister's 360 will she be able to play them? ...

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