Question

A penny, a nickel, and a dime are shaken together and thrown. Suppose that the nickel turns up heads twice as frequently as the penny and that the dime turns up heads half as frequently as the nickel. Define a sample space, and using the frequency interpretation, assign a reasonable probability density function based on the assumption that the penny is a fair coin.

Short answer

Sample space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Probability density function: HHH=0.25, HHT=0.25, HTH=0.25, HTT=0.25.

Step-by-step solution

Define the Sample Space

The three coins are the penny (P), nickel (N), and dime (D). Each coin can land on heads (H) or tails (T). The sample space consists of all possible outcomes of tossing these three coins. Therefore, the sample space is: \( S = \{ \text{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} \} \).

Determine Probability of Each Coin Landing Heads

According to the problem, the nickel lands heads twice as frequently as the penny, and the dime lands heads half as frequently as the nickel. Assuming the penny is a fair coin, it has a probability of 0.5 for heads and 0.5 for tails. Thus, the probability for the nickel landing heads is 1 (since twice 0.5), and the probability for the dime of landing heads is 0.5 (half of the nickel's probability).

Assign Probability to Each Outcome

Given the probabilities from step 2, calculate the probability for each outcome in the sample space. If we assign the probability of each coin landing with heads as follows: penny = 0.5, nickel = 1, and dime = 0.5, the probability for a specific outcome can be calculated by multiplying the probabilities together according to the heads/tails conditions of each coin. For example, the probability for HHH is \(0.5 \times 1 \times 0.5 = 0.25\). Repeat for each outcome.

Construct Probability Density Function

The probability density function assigns a probability to each of the outcomes in the sample space based on calculated values. Each outcome probability is: HHH = 0.25, HHT = 0.25, HTH = 0.25, HTT = 0.25, THH = 0.0, THT = 0.0, TTH = 0.0, TTT = 0.0. Make sure these add up to 1, consistent with a valid probability distribution.

Key concepts

Sample Space

In probability theory, the concept of a sample space is foundational. It refers to the set of all possible outcomes of a random experiment. For tossing three coins—our penny, nickel, and dime—each can land in one of two states: heads (H) or tails (T). Therefore, the sample space includes all potential combinations of these states for the three coins.
In mathematical terms, if you have three coins, the sample space is expressed as follows:

• HHH: All three coins show heads.
• HHT: The penny and nickel show heads, and the dime shows tails.
• HTH: The penny and dime show heads, and the nickel shows tails.
• HTT: The penny shows heads, and both the nickel and dime show tails.
• THH: The nickel and dime show heads, and the penny shows tails.
• THT: The nickel shows heads, and both the penny and dime show tails.
• TTH: The dime shows heads, and both the penny and nickel show tails.
• TTT: All three coins show tails.

Each possibility forms an outcome in the sample space, which is crucial for calculating probabilities.

Fair Coin

When we talk about a fair coin, it means that the coin is unbiased. This means each side of the coin—heads or tails—has an equal chance of landing up. Typically, the probability of a fair coin landing on heads is 0.5, and similarly, the probability of tails is also 0.5.
The exercise assumes the penny is a fair coin. This implies:

• The probability of the penny showing heads = 0.5.
• The probability of the penny showing tails = 0.5.

In this exercise, given that the probabilities for the nickel and dime are modified based on how frequently they show heads compared to the penny, understanding that the penny is a fair coin helps us easily establish a baseline for these calculations.

Discrete Probability Distribution

A discrete probability distribution is used to describe the probability of occurrence of each possible outcome in a sample space when the outcomes are finite and distinct.
In this scenario, after determining the unique probabilities for each coin showing heads or tails, you calculate the probability for each outcome in the sample space. How are these probabilities determined?

• The probability of each outcome is the product of the probabilities of each coin in that outcome.
• For instance, for the outcome HHH, the probability is: \(0.5 \times 1 \times 0.5 = 0.25\)
• This process is repeated for each of the outcomes in the sample space.

The result is a discrete probability distribution that assigns a probability to each outcome, ensuring the sum of all probabilities equals 1. This confirms it's a valid probability distribution.