Coin Toss Probability Formula: Definition, Examples, Solution

The Coin Toss Probability Formula is a mathematical expression used to calculate the likelihood of getting either heads or tails in a coin toss. To understand this formula better, it's essential to first grasp the concept of probability, which is a branch of mathematics that quantifies the likelihood of an event occurring. Probability is defined as the possibility of an event happening and always falls within the range of 0 (impossible event) to 1 (certain event).

Now, let's delve into the details of the coin toss probability formula and explore some examples. In this article, we'll also examine an unbiased coin, which has an equal chance of landing on either heads or tails.

Coin Toss Probability Definition

The Coin Toss Probability Formula is a mathematical expression used to determine the probability in coin toss experiments. When conducting experiments involving the tossing of two or more coins, this formula is employed to calculate the likelihood of getting heads or tails. The coin toss formula closely resembles the general probability formula, and it can be expressed as follows:

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

In the context of a coin toss experiment, the total outcomes represent all possible results of the experiment. For instance, if we are tossing two coins, the total outcomes of the coin toss experiment would include {(H, H), (H, T), (T, H), (T, T)}.

On the other hand, the favourable outcomes are the specific outcomes we desire. For example, if we want to obtain two heads when tossing two coins, the favourable outcome would be {(H, H)}.

Coin Toss Probability

When a coin is tossed, there are only two possible outcomes: a Head or a Tail. So, based on the probability formula mentioned earlier, the coin toss probability formula can be expressed as:

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

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

Therefore, the total number of possible outcomes = 2.

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

Results of Coin Toss Probability

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

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

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

When tossing a coin, the probability of getting a tail is:

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

2 Coin Toss Probability

If we toss two coins, the sample space of the event is as follows:

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

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

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Examples

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

Solution:

The required case in a two-coin toss is:

A = {(H, H)}

=> n(A) = 1

The total sample space "S" = {(H, H), (H, T), (T, H), (T, T)}

=> n(S) = 4

Probability of getting exactly two heads = P(A) = (Number of Favorable Cases) / (Total Possible Cases)

P(A) = 1/4

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

3 Coin Toss Probability

If we toss three coins, the sample space of the event is as follows:

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)}. An example based on the above sample space is as follows:

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

Solution:

The required case in a three-coin toss is:

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

=> n(A) = 3

The 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) = (Number of Favorable Cases) / (Total Possible Cases)

P(A) = 3/8

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

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Solved Examples Using Coin Toss Probability Formulas

Example 1: Calculate the probability of obtaining a head when tossing a coin.

Solution:

Total Possible Outcomes in Coin Toss = {H, T} (2 outcomes)

Favorable Outcome = {H} (1 outcome)

Probability = Favorable Outcome / Total Outcomes

P(H) = 1/2 = 0.5

Therefore, there is a 50% chance of getting a head when tossing a coin.

Example 2: Determine the probability of getting at least 1 tail when tossing two coins.

Solution:

Let's define event B as the occurrence of at least 1 tail when two coins are tossed.

Total Possible Outcomes in a two-coin toss = {(H, T), (T, H), (T, T), (H, H)} = 4 outcomes

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

Probability of Getting at least 1 tail in a 2-coin toss = P(B)

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

P(B) = 3/4 = 0.75

Thus, there is a 75% chance of getting at least 1 tail when two coins are tossed.

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Example 3: Find the probability of obtaining both a head and a tail simultaneously when tossing a single coin.

Solution:

The possible outcomes of a coin toss are {H, T}.

It is evident that there is no outcome where both Head and Tail are achieved simultaneously.

Hence, the probability of getting both a head and a tail simultaneously is zero.

Example 4: Calculate the probability of getting three heads when simultaneously tossing 3 coins.

Solution:

Let event E represent the occurrence of three heads when tossing 3 coins.

Total Possible Outcomes of three coin tosses ({HHH}, {HHT}, {HTH}, {THH}, {HTT}, {TTH}, {THT}, {TTT}) = 8 outcomes

Favorable Outcome = {HHH}

Number of Favorable Outcomes = 1 outcome

Using the Coin Toss Probability Formula:

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

P(E) = 1/8 = 0.125

Hence, there is a 12.5% chance of getting three heads when simultaneously tossing 3 coins.

Example 5: Find the probability of getting at least two heads when simultaneously tossing 3 coins.

Solution:

Let event F represent the occurrence of at least two heads when tossing 3 coins.

Total Possible Outcomes of three coin tosses ({HHH}, {HHT}, {HTH}, {THH}, {HTT}, {TTH}, {THT}, {TTT}) = 8 outcomes

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

Number of Favorable Outcomes = 4 outcomes

Using the Coin Toss Probability Formula:

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

P(F) = 4/8 = 1/2 = 0.5

Thus, there is a 50% chance of getting at least two heads when simultaneously tossing 3 coins.