# Tossing a Coin

There are always two sides to a coin- Heads and Tails.

Have you ever made an important decision by flipping a coin?

In American football, the captain of each team chooses a side of a coin and then the referee tosses the coin.

The captain who predicts the toss correctly decides about the goal which his team would defend.

Let us see how this works and what is the probability of each side of a coin.

**Lesson Plan **

**What Are the Possible Events that Can Occur When A Coin Is Tossed?**

**Every coin has two sides: Head and Tail**

We denote Head as H and Tail as Tail.

When a coin is tossed, either head or tail shows up.

The set of all possible outcomes of a random experiment is known as its **sample space**. Thus, if your random experiment is tossing a coin, then the sample space is {Head, Tail}, or more succinctly, {*H*, *T*}.

If the coin is ** fair**, which means that no outcome is particularly preferred, or every outcome is

**, then we know that for a large number of tosses, the number of Heads and the number of Tails should be roughly equal. That is, the number of Heads should be roughly 1/2 of the total number of tosses, and so should be the number of Tails. This numerical quantity of 1/2 can be used as a**

*equally likely***, or**

*measure of likelihood**probability*.

**What Do You Mean by Tossing A Coin Probability?**

**Tossing A Coin Probability is the chance of each side of the coin to show up.**

The action of tossing a coin has two possible outcomes: Head or Tail. You don’t know which outcome you will obtain on a particular toss, but you do know that it will be either Head or Tail (we rule out the possibility of the coin landing on its edge!).

Contrast this with a science experiment. For example, if your experiment is to drop an object, you know the outcome for sure: the object will fall towards the ground. However, tossing a coin is a **random experiment**, as you do know the set of outcomes, but you do not know the exact outcome for a particular execution of the random experiment.

The general formula to determine the probability is:

\(\text{Probability }= \dfrac{\text{Number of favorable Outcomes}}{\text{Total number of outcomes}}\)

When a coin is tossed, there are only two possible outcomes.

Therefore, using the probability formula

On tossing a coin, the probability of getting a head is:

P(Head) = P(H) = 1/2

Similarly, on tossing a coin, the probability of getting a tail is:

P(Tail) = P(T) = 1/2

Try tossing a coin below by clicking on the 'Flip coin' button and check your outcomes.

Click on the 'Reset' button to start again.

**How Do You Predict Heads or Tails?**

- If a coin is fair (unbiased), that is, no outcome is particularly preferred, then we cannot predict heads or tails. Both the outcomes are equally likely to show up.
- If a coin is unfair (biased), that is, an outcome is preferred, then we can predict the outcome by choosing the side which has a higher probability.
- If the probability of a head showing up is greater than 1/2, then we can predict the next outcome to be a head.
- If the probability of a tail showing up is greater than 1/2, then we can predict the next outcome to be a tail.

Suppose that you toss a coin 100 times. Which of the following results is *more likely* to occur?

**Result-1:** You obtain 95 Heads and 5 Tails

**Result-2:** You obtain 48 Heads and 52 Tails

If the coin is a *normal* everyday coin, in which neither side is particularly prone to showing up more than the other side, you would expect that in a large number of tosses, Heads and Tails should show up roughly an equal number of times. This means that out of the two results above, Result-2 seems to be the more likely one, as the number of Heads is roughly equal to the number of Tails, which concurs with the fact that neither Head nor Tail is a preferred outcome.

On the other hand, in Result-1, the number of Heads is much larger than the number of Tails. Clearly, such a result is extremely biased towards Heads, which is not very likely given that Heads and Tails are equally preferred outcomes. Note that we are not saying that Result-1 is impossible. We are only saying that it is *improbable*, or *unlikely*. In other words, the *likelihood* of Result-1 is much lower than the *likelihood* of Result-2.

The study of Probability enables us to quantify likelihoods. It enables us to answer questions like: *How likely* is Result-2? *How unlikely* is Result-1? And so on.

However, tossing a coin is a **random experiment**, as you do know the set of outcomes, but you do not know the exact outcome for a particular execution of the random experiment.

**Solved Examples**

Example 1 |

A coin is tossed a certain number of times. The relative occurrence of Heads is 0.75. Can we say that the coin is biased towards Heads?

**Solution**

No, we cannot, because the experiment (tossing the coin) may have been repeated a very small number of times, and thus the relative occurrence in such a scenario will not give the true probability.

No |

Example 2 |

Coin-A is tossed 200 times, and the relative occurrence of Tails is 0.47. Coin-B is tossed an unknown number of times, but it is known that the relative occurrence of Heads is 0.50. Which coin is fairer?

**Solution**

It is not possible to comment on the fairness of Coin-B, because the number of times it was tossed is not known. On the other hand, Coin-A seems to be fair, as the relative occurrence of tails over a large number of tosses is almost 1/2.

Coin-A |

Example 3 |

On tossing a coin twice, what is the probability of getting only one tail?

**Solution:**

On tossing a coin twice, the possible outcomes are {HH, TT, HT, TH}

Therefore, the total number of outcomes is 4

Getting only one tail includes {HT, TH}

Therefore, the number of favorable outcomes is 2

Hence, the probability of getting exactly one tail is 2/4 = 1/2

1/2 |

**Relative occurrence**- On tossing a coin, the probability of each outcome is 1/2
- P(Head) + P(Tail) = 1

**Interactive Questions **

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

- A fair coin is tossed repeatedly. If the tail appears on the first four tosses, then what is the probability of the head appearing on the fifth toss?
- If a coin is tossed thrice successively, what is the probability of obtaining at least one head and at least one tail?

**Let's Summarize **

We hope you enjoyed learning about Tossing A Coin with the simulations and practice questions. Now you will be able to easily solve problems on Tossing A Coin math with multiple math examples you learned today.

**About Cuemath**** **

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions (FAQs)**

### 1. Can a coin land on its side?

A coin can land on its side if it falls against an object such as a box, shoe, etc.

It is unlikely for a coin to land on its side on a flat surface, but we cannot say that it is impossible.

### 2. Is flipping a coin a good way to make a decision?

Making decisions by flipping a coin helps a person to decide when stuck between two options because each outcome has an equal probability.

### 3. How do you predict a coin flip?

Tossing a coin is a random experiment, as you do know the set of outcomes, but you do not know the exact outcome for a particular execution of the random experiment.

Therefore, we cannot predict a coin flip if the coin is fair.

### 4. Are coin flips truly random?

Tossing a coin is considered a random event.

According to Newton, if a person flips in a particular manner at a definite speed, then the outcome can be determined.

### 5. What is the probability of flipping five heads in a row?

On tossing a coin five times, the number of possible outcomes is 2^{5}

Therefore, the probability of getting five heads in a row is 1/2^{5}

### 6. Is flipping a coin a simple random sample?

Simple Random Sample takes a small portion of a large dataset to represent the data.

Tossing a coin is a random experiment and each outcome has equal probability.

Therefore, tossing a coin is a simple random sample.

### 7. Is a coin toss really 50/50?

On tossing a coin, each outcome has an equal probability and there are two outcomes.

Therefore, tossing a coin a 50/50.

### 8. In a coin toss, is it fairer to catch a coin or let it fall?

On tossing a coin, it is fairer to let the coin fall than catching it because the force of the hands can flip it.

### 9. What are the odds of flipping three heads in a row?

On tossing a coin three times, the number of possible outcomes is 2^{3}

Therefore, the probability of getting five heads in a row is 1/2^{3}