Question

A nickel, two dimes, and three quarters are in a cup. You draw three coins, one at a time, without replacement. What is the probability that the first coin is a nickel? What is the probability that the second coin is a nickel? What is the probability that the third coin is a nickel?

Short answer

Probabilities: First nickel: \( \frac{1}{6} \), Second if necessary: \( \frac{1}{5} \), Third: \( \frac{1}{4} \).

Step-by-step solution

Understanding the Initial Setup

We start with six coins comprising one nickel, two dimes, and three quarters. A nickel has a total value of 5 cents, a dime has 10 cents, and a quarter has 25 cents. The task is to understand the probability of drawing a nickel on the first, second, and third draws, noting that the draw is without replacement.

Calculating Probability of First Draw Being a Nickel

When drawing the first coin, there is one nickel out of the six coins in the cup. The probability that the first coin drawn is a nickel is calculated by dividing the number of nickels by the total number of coins: \( \frac{1}{6} \).

Calculating Probability of Second Draw Being a Nickel

For the second draw, consider two situations: if the first draw was a nickel (then, the probability becomes zero for the second), or if it wasn't. As the exercise doesn't specify either case, we'll assume all scenarios. Initially, if the first draw was not the nickel, then we have one nickel remaining out of five coins. The probability, in this case, would be \( \frac{1}{5} \).

Calculating Probability of Third Draw Being a Nickel

For the third draw, let's assume the nickel wasn't drawn in the first two draws (as any draw without replacement reduces the pool). Therefore, there would still be one nickel out of four remaining coins, making the probability \( \frac{1}{4} \). Additionally, if after all draws it wasn't drawn, then the distribution doesn't account for previous sequences.

Summarizing and Concluding

The process assumes that the draws are independent within permissive default draws which involves replacement depending scenarios. The first, second, and third draw have their respective probabilities: First draw as \( \frac{1}{6} \), second draw as \( \frac{1}{5} \) (if not considered from first) but one may happen earlier, and the third draw as \( \frac{1}{4} \) from earlier assumption for instance.

Key concepts

Probability without Replacement

When working with probability problems, one key aspect to consider is whether events occur with or without replacement. In problems involving probability without replacement, like the coin problem we are analyzing, the number of possible outcomes changes after each event. Initially, there are six coins in the cup. As coins are drawn and not replaced, the total number of coins decreases, which in turn affects the likelihood of drawing a specific type of coin on each subsequent draw.
This concept is crucial because as coins are removed, the probability calculations need to account for the diminishing sample space. For example, in the first draw, there is a straightforward probability since there is 1 nickel out of a total of 6 coins, calculated as \( \frac{1}{6} \).
With subsequent draws, the probability calculations change based on what has occurred before. If a nickel is drawn first, it is no longer possible to select it in the next draw, making the probability of drawing a nickel zero for that sequence of draws.

• First draw probability: \( \frac{1}{6} \).
• Second draw probability depends on the first draw outcome.
• Probabilities adjust as the sample size decreases.

This dynamic is vital to understand when analyzing sequences of dependent events, as with non-replacement scenarios, each draw impacts the next.

Coin Probability Problems

Coin probability problems often involve calculating the chances of selecting a specific type of coin from a mixed group. They are a classic example of discrete mathematics in action, requiring careful analysis of probabilities at each step. In our example, there are one nickel, two dimes, and three quarters.
Coin problems are a useful way to become familiar with applying probability concepts, like calculating probabilities under changing conditions.
To solve such problems:

• Calculate the probability for each stage independently.
• Recognize how the removal of coins impacts probabilities.
• Understand the concept of exclusivity in draws (e.g., once a nickel is drawn, subsequent draws must account for that outcome).

For example, once you've established the probability of drawing a nickel first is \( \frac{1}{6} \), you need to adjust for the second draw based on whether a nickel was taken or not. If not taken, only 5 coins are left, maintaining one nickel, leading to a probability of \( \frac{1}{5} \). Coin problems like these illustrate the essentials of probability calculations while demonstrating the impact of removing potential outcomes.

Sequential Probability Analysis

Sequential probability analysis involves evaluating the likelihood of each event within a sequence of actions, where each event is influenced by the previous ones. It's especially significant in cases without replacement, as seen in drawing coins from the cup.
This type of analysis requires you to carefully trace each step in the sequence to update your probabilities. For instance, understanding how each draw from our cup of coins changes the pool of options available is a classic example of how sequential probability works.

• Each sequential event affects the subsequent probabilities.
• It requires a recalibration of the sample space after each event.
• Initial conditions influence the entire probability chain.

In our scenario, after the first draw, the options and probabilities shift. Drawing a nickel first means adjusting the sample space to exclude it for future draws, whereas missing it means it remains possible for later draws, adjusting the probability calculations accordingly.
Thus, sequential probability analysis is key to making accurate assessments of overall probabilities and helps in building logical, step-by-step evaluations of complex problems. It highlights the importance of methodical approaches, ensuring each action's consequences are appropriately understood and factored into future predictions.