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# Coin Flip Probability Calculator

## Overview

Calculate the probability of obtaining a fixed number of heads or tails from a fixed number of tosses. Also calculate the probability of getting at least or at most a certain amount of heads or tails from a set number of tosses.

## Classical Probability

The probability of some event happening is a numerical representation of how likely that event is to occur. A probability of 1 denotes that the event will always happen 100% of the time while a probability of 0 means that the event will never happen. Classical probability entails finding out how often one outcome occurs versus another outcome and how one event occuring affects the probability of future events from happening.

Using this method: probability = (number of successful results) / (number of all possible results)

As you scale up the number of possible results or scale down the number of successful results then the probability of that event occuring decrease.

Conversely if you scale down the number of possible results or scale up the number of successful results the probability of that event occuring will increase.

## How to Calculate Probability

To calculate probability first start out by finding out all the base possibilities that can occur. In the case of a coin toss its two possibilities heads or tails.

Next determine how many times you are going to repeat the process. In the case of coin flips this would mean how many times do you want to flip the coin.

Next determine what you want to achieve. Do you want a specific outcome or at least or at most a certain amount of the same outcomes. In the case of a coin toss do you want exactly or at least or at most a certain number of heads or tails.

Use these metrics to determine how many possible successful results can exist with these criteria and how many results in total can exist.

Finally divide these two metrics to get the probability.

## Formula

The formula for getting exactly X coins from n flips is P(X) = ^{n!}⁄_{(n-X)!X!}×p^{X}×q^{(n-X)}

Where n! is a factorial which means 1×2×3×...×(n-2)×(n-1)×n.

n is the exact number of flips.

X is the exact amount of times you want to land on heads.

p is the probability of landing on heads.

q is the probability of landing on tails.

To calculate the chance of success multiply the probability by 100.

To calculate the probability as 1 in how many flips: 1 / probability.

## What is the Probability of Exactly 3 Heads

To calculate the probability of getting exactly 3 heads in 5 tosses with a probability of heads of 0.5 we will use the formula above.

In this case: n which represents the number of flips is 5.

X is the amount of times we want to land on heads which in this case is 3.

p is the probability of landing on heads which is 0.5.

q is the probability of landing on tails which in this situation is (1 - 0.5) = 0.5

Using the formula above: P(3) =

^{5!}⁄_{(5-3)!3!}×(0.5)^{3}×(0.5)^{(5-3)}P(3) =

^{120}⁄_{(2)!3!}×(0.5)^{3}×(0.5)^{(2)}P(3) =

^{120}⁄_{(2)(6)}×(0.125)×(0.25)P(3) =

^{120}⁄_{12}×0.03125P(3) = 10×0.03125

P(3) = 0.3125

Therefore the probability of getting exactly 3 heads in 5 tosses with a probability of heads of 0.5 is: 0.3125.

The chance of success = 0.3125×100 = 31.25%.

The probability 1 in is (1 / 0.3125) = 3.2. In other words you have a 1 in: 3.2 chance of getting exactly 3 heads in 5 tosses.

## What is the Probability of at Least 3 Heads

To calculate the probability of getting at least 3 heads in 5 tosses with a probability of heads of 0.5 you use the above formula once again albeit in a different manner.

In this situation you use the formula to calculate the probabilities of getting exactly 3, 4 and 5 heads and add them together. The sum of these probabilities is the probability of getting at least 3 heads.

In this case the probability of getting at least 3 heads is: P(3) + P(4) + P(5)

Using the formula above:

^{5!}⁄_{(5-3)!3!}×(0.5)^{3}×(0.5)^{(5-3)}+^{5!}⁄_{(5-4)!4!}×(0.5)^{4}×(0.5)^{(5-4)}+^{5!}⁄_{(5-5)!5!}×(0.5)^{3}×(0.5)^{(5-5)}At least 3 heads = (0.3125) + (0.15625) + (0.03125)

At least 3 heads = 0.5

Therefore the probability of getting at least 3 heads in 5 tosses with a probability of heads of 0.5 is: 0.5

The chance of success = 0.5×100 = 50%.

To calculate the probability as 1 in some number divide 1 by the probability of that event occurring.

The probability 1 in is (1 / 0.5) = 2. In other words you have a 1 in: 2 chance of getting at least 3 heads in 5 tosses.

## What is the Probability of at Most 3 Heads

To calculate the probability of getting at most 3 heads in 5 tosses with a probability of heads of 0.5 you use the above formula once again in another manner.

In this situation you use the formula to calculate the probabilities of getting exactly 0, 1, 2 and 3 heads and add them together. The sum of these probabilities is the probability of getting at most 3 heads.

In this case the probability of getting at most 3 heads is: P(0) + P(1) + P(2) + P(3)

Using the formula above:

^{5!}⁄_{(5-0)!0!}×(0.5)^{0}×(0.5)^{(5-0)}+^{5!}⁄_{(5-1)!1!}×(0.5)^{1}×(0.5)^{(5-1)}+^{5!}⁄_{(5-2)!2!}×(0.5)^{2}×(0.5)^{(5-2)}+^{5!}⁄_{(5-3)!3!}×(0.5)^{3}×(0.5)^{(5-3)}At most 3 heads = (0.03125) + (0.15625) + (0.3125) + (0.3125)

At most 3 heads = 0.8125

Therefore the probability of getting at most 3 heads in 5 tosses with a probability of heads of 0.5 is: 0.8125

The chance of success = 0.8125×100 = 81.25%.

The probability 1 in is (1 / 0.8125) = 1.2307692307692308. In other words you have about a 1 in: 1.23 chance of getting at most 3 heads in 5 tosses.

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