Flipping a Coin

What is the sample space when flipping a coin? List the event of getting heads.

Identify the experiment: We are flipping a single coin once = Coin flip experiment

List all possible outcomes: A coin can land on heads (H) or tails (T) = $S = \{H, T\}$

Define the event: The event 'getting heads' contains only one outcome = $E = \{H\}$

Answer: Sample space: $S = \{H, T\}$. Event of getting heads: $E = \{H\}$

Rolling a Die

A standard die is rolled. Find the sample space and the event of rolling a number greater than 4.

Identify the experiment: Rolling a standard 6-sided die = Die roll experiment

List all possible outcomes: A die has faces numbered 1 through 6 = $S = \{1, 2, 3, 4, 5, 6\}$

Identify outcomes in the event: Numbers greater than 4 are 5 and 6 = $E = \{5, 6\}$

Count the outcomes: Sample space has 6 outcomes, event has 2 outcomes = $n(S) = 6$, $n(E) = 2$

Answer: Sample space: $S = \{1, 2, 3, 4, 5, 6\}$. Event $E = \{5, 6\}$ with 2 favorable outcomes.

Drawing a Card

A card is drawn from a standard deck. Define the event of drawing a red ace.

Understand the sample space: A standard deck has 52 cards: 4 suits with 13 cards each = $n(S) = 52$

Identify the suits: Red suits are hearts and diamonds; black suits are clubs and spades = 26 red cards, 26 black cards

Find red aces: Red aces are: Ace of Hearts and Ace of Diamonds = $E = \{A\heartsuit, A\diamondsuit\}$

Count the event: There are exactly 2 red aces in the deck = $n(E) = 2$

Answer: The event of drawing a red ace is $E = \{A\heartsuit, A\diamondsuit\}$ with 2 outcomes out of 52 possible.

Flipping Two Coins

List the sample space when flipping two coins. Find the event of getting at least one head.

Identify the experiment: Flipping two coins (or one coin twice) = Two-coin experiment

Use systematic listing: First coin: H or T, Second coin: H or T = Combine all possibilities

List all outcomes: HH, HT, TH, TT (4 outcomes total) = $S = \{HH, HT, TH, TT\}$

Find outcomes with at least one H: At least one H means 1 or 2 heads: HH, HT, TH = $E = \{HH, HT, TH\}$

Answer: Sample space: $S = \{HH, HT, TH, TT\}$. Event of at least one head: $E = \{HH, HT, TH\}$ with 3 outcomes.

Compound Experiment

A coin is flipped and a die is rolled. How many outcomes are in the sample space? List the event of getting heads and an even number.

Count outcomes for each part: Coin: 2 outcomes (H, T). Die: 6 outcomes (1-6) = 2 and 6 outcomes

Use multiplication principle: Total outcomes = $2 \times 6 = 12$ = $n(S) = 12$

List the sample space: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 = 12 outcomes

Find favorable outcomes: Heads AND even number: H2, H4, H6 = $E = \{H2, H4, H6\}$

Answer: Sample space has 12 outcomes. Event $E = \{H2, H4, H6\}$ with 3 favorable outcomes.

Mistake: Forgetting outcomes when listing the sample space

Why: When listing outcomes manually, it's easy to miss some. For example, forgetting TH when flipping two coins.

Correct: Use a systematic approach: tree diagrams, tables, or organized lists to ensure you capture all outcomes.

Mistake: Confusing the sample space with an event

Why: The sample space contains ALL possible outcomes, while an event is a SUBSET of outcomes we care about.

Correct: Remember: $S$ = everything that CAN happen; $E$ = outcomes that match a specific condition.

Mistake: Treating different orderings as the same outcome

Why: When flipping two coins, HT and TH are different outcomes (first coin vs second coin).

Correct: Unless the problem says otherwise, treat the order as important. HT means first coin heads, second coin tails.

Quality Control

Factories inspect products and track outcomes like 'defective' or 'non-defective' to ensure quality.

A light bulb factory tests 3 bulbs. Sample space: $\{DDD, DDN, DND, NDD, DNN, NDN, NND, NNN\}$ where D=defective, N=non-defective.

Weather Forecasting

Meteorologists consider all possible weather conditions to make accurate predictions.

For tomorrow's weather, outcomes might be: sunny, cloudy, rainy, or stormy. This is the sample space for weather prediction.

Q: What is the difference between an outcome and an event?

A: An outcome is a single possible result (like rolling a 3). An event is a collection of one or more outcomes (like rolling an odd number, which includes 1, 3, and 5).

Q: Can an event be the entire sample space?

A: Yes! This is called a 'certain event' because it will definitely happen. For a die roll, the event 'rolling a number from 1 to 6' equals the entire sample space.

Q: What is an impossible event?

A: An impossible event has no outcomes - it cannot happen. For example, rolling a 7 on a standard die. We write this as $E = \{\}$ or $E = \emptyset$ (empty set).