Question

Two nickels and a dime are shaken together and thrown. All the coins are fair. We are allowed to keep the coins that turn up heads. Give two sample spaces together with probability density functions that reasonably describe this situation. Explain your answer.

Short answer

Sample spaces: $S_1=\{HHH,HHT,...,TTT\}$ and $S_2=\{0H,1H,2H,3H\}$ with uniform and derived probabilities respectively.

Step-by-step solution

Understanding the Problem

We're dealing with two types of coins: nickels and dimes. A nickel has a 50% chance of landing on heads or tails, as does a dime. We need to create sample spaces for the outcomes of shaking and throwing these three coins.

Defining Sample Space for Coin Outcomes

Each coin (the two nickels and one dime) can independently be a head (H) or a tail (T). Thus, the sample space for the outcomes of throwing three coins is:$$S_1=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}$$There are 8 possible outcomes, as each coin has 2 possible outcomes (2 x 2 x 2 = 8).

Defining Probability Density for Coin Outcomes

Each outcome in the sample space $S_1$ has a probability of occurring. Assuming all coins are fair, each outcome is equally probable. Thus, each has a probability of:$$P(x)=\frac{1}{8}$$for each x in $S_1$.

Construct Sample Space Based on 'Coins Kept'

Now let's construct a second sample space based on the number of coins (nickels or dimes) that turn up heads and can therefore be 'kept'. This sample space $S_2$ consists of the possibilities:$$S_2=\{0H,1H,2H,3H\}$$representing keeping zero to three heads.

Define Probability Density for 'Coins Kept'

Calculate the probabilities for $S_2$:- 0H (0 heads): corresponds to outcome TTT, with probability $\frac{1}{8}$.- 1H (1 head): corresponds to HHH, HHT, HTH, and THH, totaling 3 outcomes with combined probability $\frac{3}{8}$.- 2H (2 heads): corresponds to HTT, THT, and TTH, also totaling 3 outcomes with combined probability $\frac{3}{8}$.- 3H (3 heads): corresponds to outcome HHH, with probability $\frac{1}{8}$.So, $P(0H)=\frac{1}{8}$, $P(1H)=\frac{3}{8}$, $P(2H)=\frac{3}{8}$, $P(3H)=\frac{1}{8}$.

Key concepts

Sample Space

A sample space is a fundamental concept in probability theory. It is a set which includes all possible outcomes of a random experiment. In this case, our experiment involves tossing two nickels and one dime.
The possible outcomes for each coin are either heads (H) or tails (T). When combining all three coins, we have:

• Nickel 1: H or T
• Nickel 2: H or T
• Dime: H or T

Thus, our sample space consists of all combinations of these outcomes. This forms the set:

• $$S_1=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}$$

The sample space $S_1$ includes 8 outcomes because each coin can land in 2 ways, and $2\times 2\times 2=8$. Every time you shake and throw these coins, the outcome will be one of these combinations.

Probability Density Function

A probability density function (PDF) describes the likelihood of each outcome in the sample space. In our coin tossing example, since each outcome is equally probable and the coins are fair, we describe this likelihood as follows:
For the sample space $S_1$, each of the 8 potential outcomes occurs with the equal probability.

• Probability of each outcome $P(x)=\frac{1}{8}$

This probability reflects the fact that each sequence of heads and tails is as likely as any other.
Additionally, when considering a desired event like the number of heads (which determines the coins we "keep"), we define another sample space $S_2$.Here,

• Outcomes include keeping 0, 1, 2, or 3 heads (\{0H, 1H, 2H, 3H\}).
• Corresponding PDF: $P(0H)=\frac{1}{8},P(1H)=\frac{3}{8},P(2H)=\frac{3}{8},\text{and }P(3H)=\frac{1}{8}$

This PDF reflects the probabilities based on how many heads appear, considering the fairness and independence of each coin toss.

Independent Events

In probability and statistics, events are said to be independent when the occurrence of one event does not influence the outcome of another. This is particularly relevant in our exercise, where the coins tossed are fair and independent of each other.
For instance, the outcome of nickel 1 landing on heads or tails does not affect the outcome of nickel 2, nor does it affect the outcome of the dime. Because of this independence, the joint probability of any specific sequence of heads and tails is simply the product of their individual probabilities.

• Probability of heads on one coin $=\frac{1}{2}$
• Therefore, probability of all three coins showing heads $(HHH)=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{8}$

The concept of independent events is crucial in constructing sample spaces and calculating probability densities, allowing us to analyze how each toss independently contributes to the whole probability.