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Coin Flip Probability Calculator

Table of contents

Classical probabilityHow to calculate probability?More complex probabilitiesFAQs

Welcome to the coin flip probability calculator, where you'll have the opportunity to learn how to calculate the probability of obtaining a set number of heads (or tails) from a set number of tosses. This is one of the fundamental classical probability problems, which later developed into quite a big topic of interest in mathematics.

For example, maybe you like Batman and know of one of his many villains, Two-Face? You'd think that his name comes from the fact that half of his face is burnt, but no! (Okay, maybe a little bit.) He has a lucky coin that he always flips before doing anything. As this coin has two faces on it, his coin toss probability of getting a head is 1. Better not get on the wrong side (or face) of him!

🙋 If you're interested in the probability of runs in coin flips, visit our dedicated coin toss streak calculator.

Classical probability

The probability of some event happening is a mathematical (numerical) representation of how likely it is to happen, where a probability of 1 means that an event will always happen, while a probability of 0 means that it will never happen. Classical probability problems often need you to find how often one outcome occurs versus another and how one event happening affects the probability of future events happening. When you look at all the things that may occur, the formula (just as our coin flip probability formula) states that:

probability = (no. of successful results) / (no. of all possible results).

Take a die roll as an example. If you have a standard, 6-face die, then there are six possible outcomes, namely the numbers from 1 to 6. If it is a fair die, then the likelihood of each of these results is the same, i.e., 1 in 6 or 1 / 6. Therefore, the probability of obtaining 6 when you roll the die is 1 / 6. The probability is the same for 3. Or 2. You get the drill. If you don't believe me, take a die and roll it a few times, and note the results. Remember that the more times you repeat an experiment, the more trustworthy the results. So go on, roll it, say, a thousand times. We'll be waiting here until you get back to tell us we've been right all along. Go to the dice probability calculator if you want a shortcut.

But what if you repeat an experiment a hundred times and want to find the odds that you'll obtain a fixed result at least 20 times?

Let's look at another example. Say that you're a teenager straight out of middle school and decide that you want to meet the love of your life this year. More specifically, you want to ask ten girls out and go on a date with only four of them. One of those has got to be the one, right? The first thing you have to do in this situation is look in the mirror and rate how likely a girl is to agree to go out with you when you start talking to her. If you have problems with assessing your looks fairly, go downstairs and let your grandma tell you what a handsome young gentleman you are. So a solid 9 / 10 then.

As you only want to go on four dates, that means you only want four of your romance attempts to succeed. This has an outcome of 9 / 10. This means that you want the other six girls to reject you, which, based on your good looks, has only a 1 / 10 chance of happening (The sum of all events happening is always equal to 1, so we get this number by subtracting 9 / 10 from 1). If you multiply the probability of each event by the number of times you want it to occur, you get the chance that your scenario will come true. In this case, your odds are 210 × (9 / 10)4 × (1 / 10)6 = 0.000137781, where the 210 comes from the number of possible fours of girls among the ten that would agree. Not very likely to happen, is it? Maybe you should try being less beautiful!

How to calculate probability?

"Hey man, but girls and coins are two different things! I should know; I've seen at least one of each." Well, let me explain that these two problems are basically the same, that is, from the point of view of mathematics. Whether you want to toss a coin or ask a girl out, there are only two possibilities that can occur. In other words, if you assign the success of your experiment, be it getting tails or the girl agreeing to your proposal, to one side of the coin and the other option to the back of the coin, the coin toss probability will determine the answer. It all boils down to getting your hands on a coin that is weighted appropriately. Mathematically, we talk about the binomial probability distribution.

Let's look at a step-by-step example to see how to calculate the probability of an event using the coin toss probability calculator:

  1. Determine your experiment. What are the two possibilities that can happen? Assign heads to one of them and tails to the other.

  2. How many times are you going to repeat the experiment? Put that number as the number of flips in the calculator.

  3. What do you want to achieve? An exact number of successful tries? At least a set number of successful attempts? Or no more than a certain number of successful tries? Choose the correct option from the list.

  4. How many successful (exact, at least, or at most) attempts do you want to have? Input that number of heads.

  5. (Optional) If your heads and tails don't have the same probability of happening, set the right number in the Probability of heads field. Remember that in classical probability, the likelihood cannot be smaller than 0 or larger than 1.

  6. The coin flip probability calculator will automatically calculate the chance of your event happening.

More complex probabilities

You know how they say that money can't buy you happiness? Well, it's true that there are times when a coin is not enough if you want to count the likelihood of something happening. If your problem still falls under the umbrella of classical probability – meaning you can determine how many successful results there exist and how many possibilities there are in general – then the coin flip probability formula from the first section will work just fine.

If you are looking for your chances of winning the lottery or surviving on a desert island, then things start to get more complicated than a simple coin toss probability. (The former is, in fact, covered in Omni's lottery calculator.)


What is the formula for the probability in coin toss?

If you flip a fair coin n times, the probability of getting exactly k heads is P(X=k) = (n choose k)/2n, where:

  • (n choose k) = n! / (k! × (n-k)!); and
  • ! is the factorial, that is, n! stands for the multiplication 1 × 2 × 3 × ... (n-1) × n.

How do I compute the probability of 8 heads in 10 tosses?

To calculate the probability of 8 heads in 10 tosses:

  1. Recall the formula for the probability of exactly k heads in n tosses: P(X=k) = (n choose k)/2n.
  2. Thus, the probability of exactly 8 heads in 10 tosses is P(X=8) = (10 choose 8)/210 = 45/1024 ≈ 0.044.
  3. If you need the probability of at least 8 heads, find P(X=8) + P(X=9) + P(X=10).
  4. We have P(X=9) = 10/1024 ≈ 0.0098 and P(X=10) = 1/1024 ≈ 0.001.
  5. The answer is 0.044 + 0.0098 + 0.001 ≈ 0.0548.

What is the probability of 2 heads in 3 tosses?

If you toss a coin 3 times, the probability of at least 2 heads is 50%, while that of exactly 2 heads is 37.5%. Here's the sample space of 3 flips: {HHH, THH, HTH, HHT, HTT, THT, TTH, TTT​}. There are 8 possible outcomes. Three contain exactly two heads, so P(exactly two heads) = 3/8=37.5%. One outcome contains three heads, so P(at least two heads) = (3+1)/8 = 50%.

What is the probability of at least 1 head in 4 tosses?

The probability of at least 1 head in 4 tosses is 93.75%. To see why, observe that we have P(at least 1 heads) = 1 - P(no heads) = 1 - P(all tails) and P(all tails) = (1/2)4 = 0.0625. Therefore, P(at least 1 heads) = 1 - 0.0625 = 0.9375 = 93.75%, as claimed.

Experiment specifics

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