A bag has 3 red balls and 2 blue balls. Draw one with no other information, and the probability it's red is 3/5. But if you draw one without putting it back and then draw another, what's the probability the second ball is red? The answer is "it depends" — it depends on what you drew first.
If the first was a red ball, what's left is 2 red and 2 blue (4 total), so the probability the second is red is 2/4=1/2. On the other hand, if the first was a blue ball, all 3 red balls are still there, out of 4 total, so the probability is 3/4. A probability recalculated like this, under the condition that something has already happened, is called conditional probability.
A concept that often pairs up with conditional probability is independent trials. If conditional probability describes a situation where "the previous result affects the next probability," an independent trial is the exact opposite — "the previous result has no effect at all on the next one." Flipping a coin or rolling dice is completely independent every single time — just because you got heads a moment ago doesn't change the next probability.
A common question about repeated independent trials is "the probability of exactly k successes out of n." For example, if a player has a 70% free-throw success rate and takes 5 shots, what's the probability of exactly 3 successes? First, there are 5C3=10 possible orderings of "which shots were successful," and each of those orderings has the exact same probability, since it's always the product of 3 successes (0.7 each) and 2 failures (0.3 each). So the answer is 10×0.7³×0.3². A formula that multiplies "the number of orderings × the probability of that ordering" like this is called the probability of independent trials.
On our activity page, you can first check directly how conditional probability branches out with a tree diagram of drawing balls, then change the number of trials, number of successes, and success probability with sliders and watch how the probability distribution of independent trials changes as a bar chart.