Let's do this experiment on a large scale. Let's take 200 "sleeping beauties"; 100 are woken up seven times, and 100 are woken up only once. It happens 800 times that a sleeping beauty wakes up. 700 times it is one of the 100 that are woken up 7 times. 100 times it is one of the 100 that are woken up only once.
So if you wake up the chance is 7/8th that you are one of the 100 that are woken up 7 times. Well, that's the situation if Sleeping Beauty is given drugs so that she forgets she has woken up before.
Without the drugs, 100 times she is one of the 100 woken up only once, 100 times she is woken up for the first time out of seven times, and 600 times she knows that she woke up before. In that case, being woken up for the first time the chance are 50/50.
There is a nice strawman argument saying that when she is woken up, the probability is 8/14th that it is the first time, and 1/14th each that it is the second, third, ..., seventh time. Perfectly correct but it isn't the question that was asked. What was asked was what she should believe about the initial dice throw.
If she is allowed to make a bet for $1,000 each time she wakes up, what should she bet? Since she doesn't know how often she has been woken up, she should make the same decision each time. If she says heads, then the initial chance that heads was thrown is 50%. However, with tails she gets woken up 7 times. So saying "heads" gives her a 50% chance of winning once, but also a 50% chance of losing seven times. Saying tails gives her a 50% chance of losing once, but also a 50% chance of winning seven times.
She is woken up 7 times more often when the result is tails. So when she is woken up, the chance is much higher that the result is tails.
PS. If she is woken up the first time, and she is reliably told that she was woken up the first time, then heads and tails have an equal 50% chance.