I think the only way to demonstrate it to lay people is to rephrase it as the odds of picking the wrong one right off the bat. If it's 2/3 odds that you chose wrong, and I remove one card, did that change the odds that you already chose wrong?

The trick is focusing on the 1/3 odds of winning, and not mentioning there is an alternate (and correct) way of looking at the question - the initial odds of losing.