Hi Spencer.

Your question is worded strangely, so we can reword it as follows:

Since there are two types of rings and Lisa is taking three rings, you will certainly have a pair, but the third ring may match the pair or may not match the pair. So really we are asking, what is the
probability that Lisa does not get three of a kind?

It should be clear to you that this is just one minus the chances she does get three of a kind. Since it is easier to calculate the chances she gets three of a kind, let's use this approach.

The chances she gets three gold rings:

The first ring is chosen from 9, of which 5 are gold, so we have 5/9.
The second ring is chosen from 8 remaining rings, of which 4 are gold, so we have 4/8.
The third ring is chosen from 7 remaining rings, of which 3 are gold,so we have 3/7.
Multiply these together to get the chances that she pulls out three gold rings in a row:
5/9 x 4/8 x 3/7.

The chances she gets three silver rings:

The first ring is chosen from 9, of which 4 are silver, so we have 4/9.
The second ring is chosen from 8 remaining rings, of which 3 are silver, so we have 3/8.
The third ring is chosen from 7 remaining rings, of which 2 are silver, so we have 2/7.
Multiply these together to get the chances that she pulls out three silver rings in a row:
4/9 x 3/8 x 2/7.

What are the chances she pulls out either a gold triple or a silver triple? We add them:
( 5/9 x 4/8 x 3/7 ) + ( 4/9 x 3/8 x 2/7)

And so the probability she pulls out a pair and an odd ring is one minus this:
1 - ( ( 5/9 x 4/8 x 3/7 ) + ( 4/9 x 3/8 x 2/7) )

which, as you indicated, works out to 5/6.

Hope this helps,
Stephen La Rocque.