I think I'm going to make an argument that while he may be technically correct, in the spirit and context of the question, he's not quite correct.
If someone says "I saw the number 1 hit five out of six times" then the intended meaning is generally not that the first or sixth spins were the failure. The failure would have occurred on the 2nd, 3rd, 4th, or 5th spin. Thus, the multiplier is 4, not 6. If he had observed 1, 1, 1, 1, 1, 17, then he would have said "I saw the number 1 hit five times in a row" not "five times out of six."
But, even this isn't quite correct, for it assumes that there were only 6 spins, of which the first and sixth spin were 1s and only one of the other spins wasn't a 1. To understand this argument better, let's reduce it to something easier. What if I said "what are the odds of me flipping a coin and getting heads twice in a row?" Well, if there are ONLY 2 flips, then the odds of getting heads twice in a row is 1/4. Pretty simple. But, if I were working at a casino flipping the same coin over and over and over and over, then what are the odds of me getting heads twice in a row (at some point)? Pretty damn likely. And, don't forget, in 6 consecutive flips, "2 flips in a row" occurs 5 times, not 3 times.
Also, to correct Rio, the probability of getting the same thing 4 times in a row is not (1/38)^4. It's actually (1/38)^3. The probability of getting a particular thing 4 times in a row is to the 4th power, but if we spin it 3 times, it's going to come up with *some* number for the first spin 100 percent; of the time. 1/38 of the time, the 2nd spin matches the first spin. i.e. 2 of the same in a row is 1/38. But the probability of getting two 7s in a row is (1/38)^2