What is Sample Space for Three Coins?

S3-SA3-0286

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The sample space for three coins is the list of all possible outcomes when you toss three coins together. Each outcome shows whether a coin lands on Heads (H) or Tails (T).

Simple Example

Quick Example

Imagine you're playing a game of 'Heads or Tails' with your friends using three 10-rupee coins. The sample space is every single combination that could possibly happen, like getting three Heads (HHH) or two Heads and one Tail (HHT).

Worked Example

Step-by-Step

Let's find the sample space for tossing three coins.

STEP 1: List outcomes for the first coin. It can be Heads (H) or Tails (T).

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STEP 2: List outcomes for the first two coins. Combine the first coin's outcomes with the second coin's outcomes: HH, HT, TH, TT.

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STEP 3: Now, consider the third coin. For each outcome of the first two coins, the third coin can be H or T.

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STEP 4: Combine the outcomes from Step 2 with the third coin's possibilities:

- If the first two coins are HH, the third can be H (HHH) or T (HHT).

- If the first two coins are HT, the third can be H (HTH) or T (HTT).

- If the first two coins are TH, the third can be H (THH) or T (THT).

- If the first two coins are TT, the third can be H (TTH) or T (TTT).

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STEP 5: List all these unique combinations.

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ANSWER: The sample space for three coins is {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. There are 8 possible outcomes.

Why It Matters

Understanding sample space is crucial for predicting chances in games and experiments, just like how cricket analysts predict match outcomes. Data scientists use it to understand possible scenarios, and engineers use it to test how different parts of a system might behave.

Common Mistakes

MISTAKE: Missing some possible outcomes, especially combinations that look similar. For example, students might list HHT but forget HTH or THH. | CORRECTION: Use a systematic method like a tree diagram or listing method to ensure all unique combinations are included.

MISTAKE: Thinking the order doesn't matter, so HHT, HTH, and THH are considered the same outcome. | CORRECTION: For coins, the order of each coin's toss (e.g., first coin, second coin, third coin) usually matters, so HHT (first two Heads, third Tail) is different from HTH (first Head, second Tail, third Head).

MISTAKE: Confusing the number of outcomes (size of the sample space) with the actual list of outcomes. | CORRECTION: The size is how many items are in the list (e.g., 8), while the sample space is the actual list of {HHH, HHT, ...}.

Practice Questions

Try It Yourself

QUESTION: A family has three children. Each child can be a boy (B) or a girl (G). Write down the sample space for the gender of their three children. | ANSWER: {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG}

QUESTION: If you toss two coins and then roll a 6-sided die, how many outcomes are in the sample space? (Hint: First find outcomes for two coins, then multiply by die outcomes.) | ANSWER: 24 outcomes (HH1, HH2, ..., TT6)

QUESTION: List the outcomes from the sample space for three coins where exactly two heads appear. | ANSWER: {HHT, HTH, THH}

MCQ

Quick Quiz

What is the total number of outcomes in the sample space when you toss three coins?

The Correct Answer Is:

Each coin has 2 possible outcomes (H or T). For three coins, you multiply the possibilities: 2 x 2 x 2 = 8 total outcomes. Options A, B, and D are incorrect because they don't represent all unique combinations.

Real World Connection

In the Real World

When you use a mobile app for online gaming, the game often uses probability calculations based on sample spaces to decide things like which item you get from a 'loot box' or the chances of a certain event happening. Companies like Dream11 use these ideas to calculate player performance probabilities.

Key Vocabulary

Key Terms

SAMPLE SPACE: The set of all possible outcomes of an experiment. | OUTCOME: A single possible result of an experiment. | HEADS (H): One side of a coin. | TAILS (T): The other side of a coin. | PROBABILITY: The chance of a specific outcome happening.

What's Next

What to Learn Next

Great job understanding sample space! Next, you should explore 'Calculating Probability of an Event.' Knowing the sample space is the first crucial step to finding the probability of specific outcomes, like the chance of getting exactly two heads.