Question

Give two ways you could label the outcomes of an experiment that consists of flipping a dime three times. Determine a probability density function for each of the two ways.

Short answer

Label outcomes by sequences or number of heads; probabilities are $\frac{1}{8}$ per sequence, and $\frac{1}{8},\frac{3}{8},\frac{3}{8},\frac{1}{8}$ for heads count.

Step-by-step solution

Determine the possible outcomes for the first method

For the first method, we label each outcome by the sequence of heads (H) and tails (T). When flipping a dime three times, the possible outcomes are: HHH, HHT, HTH, THH, TTH, THT, HTT, TTT. This results in 8 possible outcomes.

Calculate the probability for the first method

Since each flip of the dime is independent and the probability of heads (H) or tails (T) is 0.5, the probability of obtaining any specific sequence, such as HHH or TTH, is ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$. Thus, the probability density function for this method assigns a probability of $\frac{1}{8}$ to each of the 8 outcomes.

Determine the possible outcomes for the second method

For the second method, label the outcome by the number of heads obtained in the three flips. The possible outcomes are 0 heads, 1 head, 2 heads, and 3 heads.

Calculate the probability for the second method

For this labeling method, we use the binomial probability distribution. The probability of obtaining k heads out of 3 flips is given by $P(k)=(\genfrac{}{}{0ex}{}{3}{k}){\left(\frac{1}{2}\right)}^k{\left(\frac{1}{2}\right)}^{3-k}$. Calculating for each k, we get:- Probability of 0 heads: $P(0)=(\genfrac{}{}{0ex}{}{3}{0}){\left(\frac{1}{2}\right)}^0{\left(\frac{1}{2}\right)}^3=\frac{1}{8}$.- Probability of 1 head: $P(1)=(\genfrac{}{}{0ex}{}{3}{1}){\left(\frac{1}{2}\right)}^1{\left(\frac{1}{2}\right)}^2=\frac{3}{8}$.- Probability of 2 heads: $P(2)=(\genfrac{}{}{0ex}{}{3}{2}){\left(\frac{1}{2}\right)}^2{\left(\frac{1}{2}\right)}^1=\frac{3}{8}$.- Probability of 3 heads: $P(3)=(\genfrac{}{}{0ex}{}{3}{3}){\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^0=\frac{1}{8}$.

Summarize the results

The first method assigns equal probability $\frac{1}{8}$ to each of the eight sequences of heads and tails. The second method, based on the number of heads observed, gives a probability distribution $P(0)=\frac{1}{8}$, $P(1)=\frac{3}{8}$, $P(2)=\frac{3}{8}$, and $P(3)=\frac{1}{8}$.

Key concepts

Binomial Probability Distribution

The binomial probability distribution is a common way to model situations where there are two possible outcomes, like flipping a coin. Each flip can either be a "head" or a "tail," making it a classic example for this distribution.

The probability distribution describes how likely any particular number of "heads" (or "successes") is in a series of flips. Mathematically, the probability of getting exactly $k$ heads in $n$ flips is given by the formula:

• $P(k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$

Where $(\genfrac{}{}{0ex}{}{n}{k})$ is the binomial coefficient, and $p$ is the probability of getting a head in a single flip.

For example, when flipping a coin three times, the number of heads can range from 0 to 3. The probabilities in this distribution, using the formula, will give:

• $P(0\text{ heads})=\frac{1}{8}$
• $P(1\text{ head})=\frac{3}{8}$
• $P(2\text{ heads})=\frac{3}{8}$
• $P(3\text{ heads})=\frac{1}{8}$

This distribution helps us understand the likelihood of different number of successes, and is crucial for problems where each trial is independent and follows the same probability.

Independent Events

In probability, independent events are those whose outcomes do not affect one another. This is like flipping a coin; the result of each flip doesn't change or alter the subsequent flips.

If you have a fair coin, each side has an equal chance of $\frac{1}{2}$ whether it's flipped once or multiple times. In mathematical terms, if event A and event B are independent, the probability of both A and B occurring is the product of their individual probabilities:

• $P(A\cap B)=P(A)\times P(B)$

Let's consider the example of flipping a dime three times. Each flip's outcome does not affect the other flips. It’s always an independent event—meaning each time the probability remains $\frac{1}{2}$ for either heads or tails.

This concept is vital, as assuming independence incorrectly can lead to wrong conclusions in probability tasks. Ensuring events are truly independent allows for accurate calculation of probabilities across multiple actions or trials.

Outcomes in Probability

Outcomes in probability refer to the possible results of a probabilistic experiment. Visualize flipping a coin: the potential outcomes are "heads" or "tails." When you flip a coin three times, the outcomes multiply into sequences such as HHH, HHT, and so forth. Each of these sequences is an outcome of that experiment.

For a set number of trials, like flipping a coin, you can count the possible outcomes with the formula $2^n$, where $n$ is the number of flips. So, flipping a coin three times yields

• 8 outcomes: HHH, HHT, HTH, THH, TTH, THT, HTT, TTT

Each specific sequence has its own probability, which can be calculated if events are independent. In our coin flip example, each outcome has a probability of ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$.

Understanding outcomes is key to calculating probability distributions. They help visualize all the possible results and assign probabilities to them, ensuring a clear view of how likely different scenarios are in experiments.