Probability of an outcome Exactly n Times
Multiple Trial Probability
To formula to calculate the probability that an event will occur exactly n times over multiple trials is intricately tied to the formula for combinations. This may be a surprise at first, but upon examination there is an clear connection between combinations and multiple trial probabilities.
Part I. How we get the formula for multiple trial Probability
If you're on this page, you hopefully know that if you flip a coin you have a ½ probability of getting heads.
But let's consider the following situation. What if we flip the coin twice. What is the probability that we will get heads both times. This is an independent event because the first event, the coin toss, does not effect the second event, the second toss.
The probability of these two independent events is ¼ !
Let's look at the sample space for these tosses

What about three independent events? What is the probability that we will get heads exactly three times if we toss the coin three times?
The probability of getting heads all three times is
Let's look at the sample space for these tosses
Three Ways that we can get 1 Heads out of three tosses
Three Ways that we can get 1 Tails out of three tosses
2 ways to get All heads or all tails
Developing the Formula
Formula for Probability of Independent Events
Formula = The general formula is to determine how many combinations of the independent events can occur, then multiply the probability of each by the result of the combination. It's much easier to understand by looking at some more examples.
A fair coin is tossed 10 times, what is the probability that it falls on heads exactly 6 times?
If 5 dice are tossed, what is the probability that they show exactly 3 fives?
