Problem 13 - Algebra and Trigonometry

Question

List the sample space S of each experiment and (b) construct a probability model for the experiment. Tossing two fair coins and then a fair die

Step-by-Step Solution

Answer

The sample space S has 24 outcomes, each with a probability of 1/24.

Step 1: Define the Experiment

The given experiment involves tossing two fair coins followed by rolling a fair die.

Step 2: Determine the Sample Space for Two Coins

Each coin has two possible outcomes: Heads (H) or Tails (T). For two coins, the sample space is: S_coins = {(H,H), (H,T), (T,H), (T,T)}

Step 3: Determine the Sample Space for One Die

A fair die has six possible outcomes: 1, 2, 3, 4, 5, 6. S_die = {1, 2, 3, 4, 5, 6}

Step 4: Combine the Two Sample Spaces

The sample space for the entire experiment is the combination of the sample space of two coins and one die. S = {((H,H),1), ((H,H),2), ((H,H),3), ((H,H),4), ((H,H),5), ((H,H),6), ((H,T),1), ((H,T),2), ((H,T),3), ((H,T),4), ((H,T),5), ((H,T),6), ((T,H),1), ((T,H),2), ((T,H),3), ((T,H),4), ((T,H),5), ((T,H),6), ((T,T),1), ((T,T),2), ((T,T),3), ((T,T),4), ((T,T),5), ((T,T),6)}

Step 5: Assign Probabilities

Since the coins and the die are fair, each outcome is equally likely. The total number of outcomes is 4 (coins) × 6 (die) = 24. Therefore, the probability of each outcome is P(each outcome) = 1/24. The probability model is: P(((H,H),1)) = 1/24, P(((H,H),2)) = 1/24, ..., P(((T,T),6)) = 1/24

Key Concepts

Probability Model

A probability model lays out all possible outcomes of an experiment and assigns probabilities to them. For our experiment, each step needs a clear understanding before we combine them all.
Firstly, tossing two fair coins has four possible outcomes: Heads-Heads (H,H), Heads-Tails (H,T), Tails-Heads (T,H), and Tails-Tails (T,T).
Next, a fair die roll has six outcomes: 1, 2, 3, 4, 5, and 6.
Combining these gives 24 possible outcomes in total.
Since all outcomes are equally likely, each has a probability of 1/24.

Combining Sample Spaces

Combining sample spaces means generating every possible outcome from multiple experiments.
For example, the first sample space, S_coins, lists outcomes from tossing two coins: {(H,H), (H,T), (T,H), (T,T)}.
The second sample space, S_die, lists outcomes from rolling a die: {1, 2, 3, 4, 5, 6}.
To combine these spaces, we create pairs of each outcome from both spaces.
Resulting in 24 different combinations like ((H,H),1), ((H,H),2),...,((T,T),6).
This showcases how different events together form a combined sample space.

Fair Coins

A fair coin is one that has an equal chance of landing heads or tails when tossed.
Each coin toss independently results in heads (H) or tails (T) with a probability of 0.5 or 50%.
When two coins are tossed, the combined sample space becomes: (H,H), (H,T), (T,H), and (T,T).
Each of these outcomes has a probability of 1/4, as they are equally likely.
This principle of 'fairness' ensures that probability calculations are straightforward and consistent.

Fair Die

A fair die is one where each face, numbered 1 through 6, has an equal chance of landing face up.
When rolled, each side has a probability of 1/6.
This uniformity simplifies the calculation of combined events.
For example, paired with our fair coins' outcomes, each combination with the die maintains equal likelihood.
Considering our 24 outcomes of coin tosses and die rolls, each pair remains equally probable because the underlying elements—the fair coins and the fair die—are unbiased.