Tossing a Coin Probability Formula
Last Updated : 03 May, 2025
Coin Toss Probability helps us to determine the likelihood of getting heads or tails while flipping a coin. Before diving into the formula, it's essential to understand that when a fair coin is tossed, there are only two possible outcomes: Heads (H) and Tails(T).
In the fair coin toss definition, each outcome has an equal chance of occurring, which means the probability of getting heads and tails is 50 %.
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Tossing a coin probability formula is the formula that is used to find the probability in a coin toss experiment. Suppose we carried out an experiment in which we tossed two or more coins, and the probability of finding heads 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,
\text{Probability} = \dfrac{Number of Favourable Outcomes}{Total Outcomes}
The total outcome of the coin toss experiment is all the outcomes of the experiment. Suppose we toss two coins; then the total outcomes of the coin toss experiment are:
{(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)}.
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Tossing a Coin Probability
If we toss a coin, then there are only 2 possible outcomes, i.e. either a Head or a Tail. The given below is the illustration of all possible outcomes when a coin is tossed three times. The tree shows all the possible combinations of heads (H) and tails (T) for three coin tosses.
All the possible outcomes when three coins are tossed togetherThe coin toss probability formula is given as,
Coin Toss Probability Formula = (Number of Favourable Outcomes)/ (Total Possible Outcomes)
If a single coin is tossed, the Total Possible outcomes are either Head(H) or Tail(T)
In a coin toss, we can have two favourable outcomes, either Head(H) or Tail(T)
Then, the total number of possible outcomes = 2
Tossing 1 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 a 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
Tossing 2 Coins 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.
Tossing 3 Coins 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.
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Solved Examples on Tossing a Coin Probability
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 a head and a 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.
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