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

First Toss Second Toss







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

If we spin the red arrow four times. What is the probability that we will spin an odd number on the fist spin and even numbers on all of the other spins?

p(odd) p(even) p(even) p(even)

To calculate the probability of independent events simply multiply each probability together

× × × = =

Imagine that we are using the same spinner depicted up above. Calculate the probability of obtaining exactly 1 odd number on 4 spins of the arrow.

        Probability
O E E E
E O E E
E E O E
E E E O

To determine the total probability of these independent events add up each fraction: . Note, this is the same as . Quick refresher on the formula for combinations in math.

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.


Problem 1

What is the probability of obtaining two odd numbers on four spins?

Problem 2

What is the probability of obtaining three odd numbers on four spins?

Problem 3

A fair coin is tossed 10 times, what is the probability that it falls on heads exactly 6 times?

Problem 4

If 5 dice are tossed, what is the probability that they show exactly 3 fives?


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