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Scientific Calc

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.
Answer


What about three independent events? What is the probability that we will get heads exactly three times if we toss the coin three times?

Answer
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 ?
Answer

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.
Answer

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.
What is the probability of obtaining two odd numbers on four spins?
Answer
What is the probability of obtaining three odd numbers on four spins?
Answer
A fair coin is tossed 10 times, what is the probability that it falls on heads exactly 6 times?
Answer
If 5 dice are tossed, what is the probability that they show exactly 3 fives?
Answer