How many possible outcomes would be there if three coins were tossed once?
Last Updated :
27 Aug, 2024
[Tex]Total Possible Outcomes = 2^3 = 8 [/Tex]
Explanation:
We will learn 2 Methods on how to solve this problem :
1) Logical Method
Step 1: First of all try to find out all the possible outcomes when a single coin is toss.
When we toss a fair coin then the outcomes are ‘Head’ or ‘Tail’
Step 2: Represent them in form of sample space.
We will represent head as ‘H’ and tail as ‘T’
Sample Space, S = { H, T}
Step 3: If the same process is repeated then try to find out the relation between them.
Here three coins are tossed, so the first coin may show Head or Tail, similarly, the second and third coins may also show head and tail.
The first coin has two possibilities, the second coin has two possibilities and the third coin also has two possibilities.
So total number of possibilities = 2 × 2 × 2 = 8
Step 4: Write down all the possibilities.
By exchanging the position of head and tail, all the possible outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
So there is a total of 8 possible outcomes when three coins were tossed once.
OR
2) Formula Method
The formula to calculate the number of possible outcomes when tossing n coins is:
[Tex]Total Possible Outcomes = 2^n[/Tex]
Explanation:
n represents the number of coins being tossed.
2 represents the two possible outcomes for each coin toss (Heads or Tails).
For example,
if you toss 3 coins, the total number of possible outcomes would be:
[Tex]Total Possible Outcomes = 2^3 = 8 [/Tex]
Concept
In everyday life, the word ‘maybe’ is often used when people are unsure about certain things. For example, perhaps, it may rain today. There could be a chance of rain but we are not sure, it might not rain today. This type of statement leads to uncertainty in the event. The term Probability is derived from the word ‘probably’, which means when people are not sure of the occurrence of an event. So the unique way of finding opportunities to be discussed in this article.
Probability
It is defined as the ratio of favourable outcomes to the total number of possible outcomes. We can represent it in fractions, decimals, or percentages. The probability lies between 0 to 1.
Probability = {(Favourable Outcomes) / (Total Outcomes)}
We generally represent it by ‘p’.
If probability of an event is ‘p’ then the probability of not happening of same event is ‘1-p’, we generally represent the non-happening of probability by ‘q’.
q = 1 – p
Difference between Probability and Possibility
Do not get confused and consider Possibilities for Probability. Probability and Possibility are two similar concepts but has different context and distinct meaning.
In short we can say, Possibility means something can happen, without specifying how likely it is. Probability measures how likely something is to happen, expressed as a number between 0 (impossible) and 1 (certain). Probability is Quantitative and can be measured While Possibility is Qualitative and cannot be measured in numbers.
Key Differences
- Scope: Possibility is about whether something can happen at all, while probability is about how likely it is to happen.
- Measurement: Possibility is binary (something is either possible or not), whereas probability is a spectrum (something can be more or less likely).
- Application: Possibility is often used in everyday language to discuss potential outcomes, while probability is used in mathematical, statistical, and scientific contexts to assess risk, likelihood, and uncertainty.
For Example,
Probability is “The chances of getting a double six (both dice showing a six) in two consecutive roll is only 1 in 36”
While Possibility is “The Chances of you, finding a Million Dollar under your bed”
In short, possibility is about potential, while probability is about likelihood.
Terms Related to Probability
- Random Experiment: Random experiment is defined as any job to be done. Suppose we are rolling dice or tossing a coin etc.
- Event: Event is defined as the collection of outcomes.
- Sample Space: Sample space is defined as all the possible outcomes of an experiment. For example, if we roll a dice then it might appear 1,2,3,4,5, or 6. So all these outcomes are termed as sample space.
Possible Outcomes
Possible outcomes are termed as all outcomes which may appear when an experiment is performed. It may be in favor of the person who is experimenting and it may also not be in favour. Suppose a person rolls a dice and he wished that 5 will appear but the final outcome may be any number between 1 to 6.
All the possible outcome together is termed as the Sample Space.
And if we count all the possible outcomes then that value is called the number of possible outcomes.
Similar Questions
Question 1: How many possible outcomes if two coins were tossed once?
Solution:
When we toss a fair coin then the outcomes as ‘Head’ or ‘Tail’, so the total possible outcomes are 2.
We will represent head as ‘H’ and tail as ‘T’
Since the same experiment is performed twice so total possible outcomes = 2 × 2
= 4
Possible outcomes, S = { HH, HT, TH, TT }
So there is a total of 4 possible outcomes when two dice were tossed once.
Question 2: How many possible outcomes would be there if four coins were tossed once?
Solution:
When we toss a fair coin then the outcomes as ‘Head’ or ‘Tail’, so the total possible outcomes are 2.
We will represent head as ‘H’ and tail as ‘T’
Since the same experiment is performed four times, so total possible outcomes = 2 × 2 × 2 × 2
=16
Possible outcomes, S = { HHHH, HHHT, HHTH, HTHH, THHH, HHTT, TTHH, HTHT, THTH, THHT, HTTH, HTTT, THTT, TTHT, TTTH, TTTT }
So there is a total of 16 possible outcomes when four coins were tossed once.
Question 3: What are the possible outcomes if you roll a die twice?
Solution:
Each time we roll a die the number of outcomes we can obtain are 6 (1,2,3,4,5,6)
So for two die , the possible number of outcomes would be :
[Tex]6^2 = 36[/Tex]
Here are all 36 possible outcomes:
- (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
- (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
- (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
- (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
- (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
- (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
So, there are 36 possible outcomes when you roll a die twice.
Question 4: How many outcomes are possible when tossing a single die and a single coin together?
Solution:
Number of Possible Outcomes = 12 (H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6)
Question 5: What are the possible outcomes if you draw two cards one after the other from a deck of cards (with replacement)?
Solution:
When we draw the 1st card from the deck of cards there are 52 possible outcomes
Now when we draw the 2nd card from the deck of cards there are again 52 possible outcomes because replacement is allowed after drawing the 1st card
So the total possible outcomes for drawing two cards from a deck of cards with replacement is:
[Tex]52² = 52 \times 52 = 2704[/Tex]
Question 6: What are the possible outcomes if you draw two cards one after the other from a deck of cards (without replacement)?
Solution:
When we draw the 1st card from the deck of cards there are 52 possible outcomes
Now when we draw the 2nd card from the deck of cards there are 51 possible outcomes because 1 card has been drawn and replacement is not allowed.
So the total possible outcomes for drawing two cards from a deck of cards without replacement is:
[Tex]52 \times 51 = 2652[/Tex]
Also Read:
Conclusion
When you toss three coins simultaneously, there are 8 possible outcomes. Each of these outcomes represents a different combination of Heads and Tails. This principle can be applied to any number of coins, where the total number of possible outcomes is calculated by raising 2 (the number of outcomes per coin) to the power of the number of coins tossed. In this case, for three coins, the calculation is 2³ = 8.
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