# Coin Toss Probability

Coin toss is nothing but the experiment of tossing a coin. When the probability of an event is same as zero, then the event is said to be impossible. When the probability is same as one then the event is said to be sure or certain. When we toss a coin we can only have two types of outcomes: heads or tails. At no point of time we can have both heads and tails as outcomes together whenever we flip a coin. We know that probabilities lie between 0 and 1. We always consider to have fair coins being tossed.

## Definition

The sample space count can be decided by the number of coins being flipped together. If we have flipped ‘n’ coins or flipped a coin ‘n’ times then the sample space in the experiment will have $2^n$ elements.

For example in case of tossing a coin thrice we can have exactly 2^3 = 8 outcomes that are HHH, HHT, HTH, THH, TTH, THT, HTT, TTT.

We know that probability of an event is the ratio of the number of outcomes associated with that event to the total number of outcomes of the experiment. Here, suppose we need to find the probability of getting all heads, then there is only one outcome possible in such a case so the probability of this event is 1/8.
Also we see that at any point of time both heads and tails cannot come as outcome simultaneously. These events have an equally likely chance and also their probability is same in the experiment of tossing a coin.

Thus the events of getting a head / a tail are mutually exclusive. It is impossible to get a head and a tail simultaneously when a coin is flipped and the event of getting either a head or a tail is a certain event.

When we add the probability of getting a head and probability of getting a tail when a coin is tossed the sum is always equal to 1.

In general when we have flipped a coin large number of times then we are making a relative frequency estimate of probability on easy basis. For example if we calculate the probability of getting 75 heads out of 100 times then we say it will be $\frac{75}{100}$ = 0.75. as we flip the coin more number of times more approximation we get.

This means rather getting 0.5 directly we will get answers approaching to 0.5 or other values. For example if we have flipped a coin 10000 times and we got 5678 times heads, then the probability is 0.5678 which is not exactly 0.6. It is simply approaching to 0.6.

## Examples

Few examples are based on coin toss probability:

Example 1: There are 128 outcomes possible in a coin toss experiment. What are the number of coins been tossed together.

Solution:

Let the number of coins be ‘n’.

We know that $2^n$ = 128

$\rightarrow$ $2^n$ = $2^7$

$\rightarrow$ n = 7

Hence, 7 coins are flipped together.

Example 2: Three coins are tossed. Find the probability of at least 2 tails.

Solution:

S = {TTT, THT, TTH, HTT, THH, HTH, HHH, HHT}

$\rightarrow$ N (S) = 8

B = At least 2 tails = {TTT, TTH, THT, HTT}

$\rightarrow$ N (B) = 4

P (B) = $\frac{4}{8}$ = $\frac{1}{2}$