Problem 6 - Algebra and Trigonometry

Question

An experiment consists of tossing a coin and drawing a card from a deck. (a) How many elements does the sample space have? (b) List the elements in the event "getting heads and an ace." (c) List the elements in the event "getting tails and a face card." (d) List the elements in the event "getting heads and a spade."

Step-by-Step Solution

Answer

(a) 104 elements. (b) 4 elements: (H, Ace of Spades), (H, Ace of Hearts), (H, Ace of Diamonds), (H, Ace of Clubs). (c) 12 elements with Tails and any face card. (d) 13 elements with Heads and any spade.

Step 1: Understanding the Experiment

The experiment involves two independent actions: tossing a coin and drawing a card from a standard deck of 52 cards. The sample space will involve all possible combinations of these two actions.

Step 2: Calculating Sample Space

Each coin toss can result in either heads (H) or tails (T), making two outcomes for the coin. Each card draw can result in one of 52 cards. Thus, the total number of elements in the sample space is given by multiplying the outcomes of each action: \[ 2 \text{ (coin outcomes) } \times 52 \text{ (card outcomes) } = 104 \text{ elements} \]

Step 3: Event 'Getting Heads and an Ace'

For this event, we need to consider only the outcomes where the coin shows heads and the card drawn is an ace. There are four aces in a deck (Ace of Spades, Ace of Hearts, Ace of Diamonds, and Ace of Clubs). Thus, the elements are: - (H, Ace of Spades) - (H, Ace of Hearts) - (H, Ace of Diamonds) - (H, Ace of Clubs)

Step 4: Event 'Getting Tails and a Face Card'

For this event, consider the outcomes where the coin shows tails and the card drawn is one of the face cards (Jack, Queen, King) in any of the four suits. There are 12 face cards in a deck. Thus, the elements are: - (T, Jack of Spades) - (T, Queen of Spades) - (T, King of Spades) - (T, Jack of Hearts) - (T, Queen of Hearts) - (T, King of Hearts) - (T, Jack of Diamonds) - (T, Queen of Diamonds) - (T, King of Diamonds) - (T, Jack of Clubs) - (T, Queen of Clubs) - (T, King of Clubs)

Step 5: Event 'Getting Heads and a Spade'

For this event, we look at outcomes where the coin shows heads and the card drawn is a spade. The spades in a deck include 13 cards (from Ace to King of Spades). Thus, the elements are: - (H, Ace of Spades) - (H, 2 of Spades) - (H, 3 of Spades) - (H, 4 of Spades) - (H, 5 of Spades) - (H, 6 of Spades) - (H, 7 of Spades) - (H, 8 of Spades) - (H, 9 of Spades) - (H, 10 of Spades) - (H, Jack of Spades) - (H, Queen of Spades) - (H, King of Spades)

Key Concepts

Sample Space

When discussing probability, a **sample space** is a critical concept. In simple terms, the sample space represents all possible outcomes of a given experiment.
In the exercise, where we toss a coin and draw a card, the sample space consists of every possible combination of coin outcomes and card draws.
Since a coin toss can result in either heads (H) or tails (T), and a standard deck has 52 different cards, each combined action contributes to the sample space.
In this case, there are 104 potential outcomes in the sample space, as calculated by multiplying the individual outcomes of the coin toss (2) by the outcomes of the card draw (52): \[ 2 \text{ outcomes (coin)} \times 52 \text{ outcomes (cards)} = 104 \text{ elements} \]Understanding the sample space helps to systematically consider every possible scenario of the experiment and is foundational for calculating the probabilities of specific events.

Combinatorics

In probability, **combinatorics** is all about counting combinations. It is especially useful when trying to understand and list events from a sample space.
Combinatorics simplifies enumerating combinations and permutations, which can otherwise be tedious.
Consider event (b) in the exercise: "getting heads and an ace."
There are four aces available in the deck. Thus, through combinatorial analysis, we can quickly determine the possible combinations:

• (H, Ace of Spades)
• (H, Ace of Hearts)
• (H, Ace of Diamonds)
• (H, Ace of Clubs)

Similarly, understanding combinatorics also aids in calculating outcomes for events like "getting tails and a face card" where there are 12 face cards in total, and "getting heads and a spade" with 13 spades available.
Combinatorics offers a systematic way to count these configurations precisely without missing any possible outcome.

Independent Events

**Independent events** are a vital concept in probability that impact how we calculate probabilities. Two events are considered independent if the outcome of one event does not affect the outcome of the other.
In the context of the exercise, tossing a coin and drawing a card are independent actions. The result of the coin toss—whether it's heads or tails—does not influence the type of card drawn from the deck.
This independence permits us to multiply the probabilities when calculating the likelihood of both events occurring together.
For instance, if you want to determine the probability of "getting heads and an ace," you would multiply the probability of getting heads with the probability of drawing an ace from the deck.
This characteristic of independence simplifies predictions and calculations involving multiple events, as the individual probabilities are directly multiplied without any adjustments for conditional dependence.