The Easy Example: A pocket pair
You start with a pair of Jacks in the pocket. Not too shabby. The flop however, doesn't contain another Jack.
Lesson 1: What's my chance of getting a Jack on the turn?
You need to just figure out the number of outs and divide it by the number of cards in the deck. There's 2 more Jacks. There's 47 more cards since you've seen five already. The answer is 2/47, or .0426, close to 4.3%.
Lesson 2: No luck on the turn, how 'bout the river?
Still 2 Jacks left, but one less card in the deck bringing the grand total to 46. What's 2/46? That's .0434, which is also close to 4.3% Your chances didn't change much.
Lesson 3: Screw getting just one Jack! I want them both! What are my chances?!
Since we're trying to figure out the chances of getting one on the turn AND the river, and not getting one on EITHER the turn or river, we don't have to reverse our thinking. Just multiply the probability of each event happening. Chances of getting that first Jack on the turn was .0426, remember? The chance of getting a second Jack on the river would be 1/46, because there'll only be one Jack left in the deck. That's about .0217, or 2.2%. To get the answer, multiply 'em. .0426 X .0217 is about .0009! That's around one-tenth of a percent. I wouldn't bank on that one.
Lesson 4: Hey, what were my chances of getting a pair of Jacks anyway?
To figure that out, think of it as getting dealt one card, then another. What are your chances of the second card matching the first one? There will be 3 cards left like the one you have. There's 51 cards left in the deck. 3/51 is .059 or 5.9%. What the chance that it'll be Jacks? Well, there's 13 different cards. So, .059/13 is about .0045, a little less than half a percent.
Lesson 5: What were my chances of getting a Jack on the flop?
Now you do have to "think in reverse" as in the previous example. Figure out the chances of NOT getting a Jack on each successive card flip. First card you have a 48/50 chance (48 non-Jack cards left, 50 cards left in the deck), second card is 47/49, third card is 46/48. Those come out to .96, .959, and .958. Multiply them and get .882, or an 88.2% chance of NOT getting any Jacks on the flop. Invert it to figure out what your chances really are and you get .118 or 11.8%. This will be your chance to get one or two Jacks.