Tossing a Coin Probability Formula

Tossing a Coin Probability: Coin Toss Probability Formula is the formula that tells us the probability of finding the head or tail in a coin toss. Before learning more about the coin toss probability formula, let’s learn more about, What is Probability? Probability is a branch of Mathematics that tells how likely an event occurs. We define it as, the possibility of happening an event. Its value always lies between 0 (zero) to 1 (one) where 0 indicates an impossible event and 1 indicates a certain event. 

Now let’s learn more about the coin toss probability formula and examples in detail in this article. The following image shows an unbiased coin that has an equal probability of landing both heads and tails.

Coin Toss Probability Formula Definition

Tossing a coin probability formula is the formula that is used to find the probability in the coin toss experiments. Suppose we carried out an experiment in which we toss two or more coins and the probability of finding the head, or tails in that experiment is calculated using the coin toss formula. The coin toss formula resembles the normal probability formula and the coin toss probability formula is,

Probability = (Number of Favourable Outcomes)/(Total Outcomes)

The total outcome of the coin toss experiment is all the outcome of the experiment, suppose we toss two coins then the total outcomes of the coin toss experiment is {(H, H), (H, T), (T, H), (H, H)}

And the favourable outcome in the outcome that we desire to suppose we want two heads in tossing two coins then the favourable outcome is, {(H, H)}

Tossing a coin probability

If we toss a coin then there are only 2 possible outcomes, i.e. either a Head or a Tail. So, as per the above probability formula, the coin toss probability formula is given as,

Coin Toss Probability Formula = (Number of Favourable Outcomes)/ (Total Possible Outcomes)

If a single coin is tossed, Total Possible Outcomes are either Head(H) or Tail(T) 

Then, the total number of possible outcomes = 2

In a coin toss, we can have two favourable outcomes either Head(H) or Tail(T)

Results of Tossing a coin probability

In a coin toss, there are only two possible outcomes. Therefore, using the coin toss probability formula:

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

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

• On tossing a coin, the probability of getting a tail is,

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

2 Tossing a coin probability

If we toss two coins then the sample space of the event is,

S = {(H, H), (H, T), (T, H), (T, T)}

Now the event of getting exactly one head is represented as, {(H, T), (T, H)}. Similarly, an example based on the above sample space is,

Example: Find the probability of getting exactly two heads when we toss two coins.

Solution:

The required case in two coin toss is,

A = {(H, H)}

=> n(A) = 1

Total sample space “S” = {(H, H), (H, T), (T, H), (T, T)}

=> n(s) =  4

Probability of getting exactly two heads = P(A) = (Favourable Case)/(Total Case)

 P(A) = 1/4

Thus, the probability of getting two heads in two coin toss is 1/4.

3 Tossing a coin probability

If we toss three coins then the sample space of the event is,

S = {(H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, T, H), (T, T, T), (T, H, H), (T, H, T)}

Now the event of getting exactly three heads is represented as, {(H, H  H), (T, H)}. Similarly, an example based on the above sample space is,

Example: Find the probability of getting exactly two heads when we toss three coins.

Solution:

The required case in two coin toss is,

A = {(H, H, T), (H, T, H), (T, H, H)}

=> n(A) = 3

Total sample space “S” = {(H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, T, H), (T, T, T), (T, H, H), (T, H, T)}

=> n(s) =  8

Probability of getting exactly two heads = P(A) = (Favourable Case)/(Total Case)

 P(A) = 3/8

Thus, the probability of getting two heads in three coin toss is 3/8.

Read More:

• Probability Theory
• Chance and Probability
• Empirical Probability

Solved Examples Using Tossing a Coin Probability Formulas

Example 1: Find the probability of getting a head when a coin is tossed.

Solution: 

Total Outcomes of Coin Toss = {H, T} (2)

Favorable Outcome = {H} (1)

Probability = Favourable Outcome/ Total Outcome

P(H) = 1/2 = 0.5

So there is a 50% chance of getting a head when a coin is tossed.

Example 2: Find the probability of getting at least 1 tail when two coins are tossed.

Solution: 

Let B be the event of getting at least 1 tail if two coins are tossed.

Total Outcomes of two coin toss = {(H, T), (T, H), (T, T), (H, H)} = 4

Number of Favorable Outcomes = {(H, T), (T, H), (T, T)} = 3

Probability of Getting at least 1 tail if 2 coins are tossed = P(B)

 P(B) = (Number of Favorable Outcomes)/(Total Possible Outcomes)

P(B) = 3/4 = 0.75

So there are 75% of chance of getting at least 1 tail when two coins are tossed.

Example 3: Find the probability of getting head and tail at the same time when a single coin is tossed.

Solution:

The outcome of a coin toss is, {H, T}

We see that there is no outcome when the Head and Tail are achieved simultaneously.

Thus, the probability of getting head and tail simultaneously is Zero.

Example 4: Find the probability of getting three heads when 3 coins are tossed at the same time.

Solution:

Let E be the event of getting three heads when 3 coins are tossed.

Total Possible Outcomes of three coin toss ({HHH}, {HHT}, {HTH}, {THH}, {HTT}, {TTH}, {THT}, {TTT})

Total Number of Possible Outcomes = 8

Favorable Outcomes = {HHH}

Number of Favorable Outcomes = 1

As per the Coin Toss Probability Formula,

P(E) = (Number of Favorable Outcomes)/(Total Number of Possible Outcomes)

P(E) = 1/8 = 0.125

So, there is a 12.5% chance of getting all 3 heads when 3 coins are tossed.

Example 5: Find the probability of getting at least two heads when 3 coins are tossed at the same time.

Solution: 

Let F be the event of getting at least two heads when 3 coins are tossed.

Total Possible Outcomes of three coin toss ({HHH}, {HHT}, {HTH}, {THH}, {HTT}, {TTH}, {THT}, {TTT})

Total Number of Possible Outcomes = 8

Favorable Outcomes = ({HHT}, {HTH}, {THH}, {HHH})

Number of Favorable Outcomes = 4

As per the Coin Toss Probability Formula,

P(F) = (Number of Favorable Outcomes)/(Total Number of Possible Outcomes)

P(F) = 4/8 
        = 1/2 = 0.5

So, there is a 50% chance of getting at least two heads when 3 coins are tossed.

FAQs on Tossing a Coin Probability Formula

Q1: What is Probability?

Probability is a branch of mathematics that studies the chances of happening an event based on the previous outcome and other factors. It is highly used in Statics, Risk Analysis, Insurance Sector and others.

Q2: What are the Possible Outcomes of a Coin Toss?

The possible outcomes of a coin toss are either the coin lands on the Head or the coin lands on the Tail. The sample space (S) of a coin toss is,

S = {H, T}

Q3: What is the Tossing a Coin Probability Formula?

The coin toss probability formula is,

P(S) = (Favourable Outcome)/ (Total Outcome)

Q4: What is the Sample Space when Two Coins are Tossed?

The sample space denoted by S when two coins are tossed is,

S =  {(H, T), (T, H), (T, T), (H, H)} 

where,
H represents the Head
T represents the Tail

Q5: What is the Probability of a Head or Tail in a Coin Toss?

There is an equal probability of getting Head{H} or Tail{T} in a coin toss. A coin toss can have two outcomes and the probability of the outcome is 0.5. If the probability of the Head is P(H) and the probability of the tail is P(T) then,

P(H) = P(T) = 0.5