Problem 56 - College Algebra

Question

Write a probability word problem whose answer is one of the following fractions: \(\frac{1}{6}\) or \(\frac{1}{4}\) or \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The probability of getting a '2' when a dice is rolled once is \(\frac{1}{6}\). The probability of picking a blue ball from a bag that contains 4 differently colored balls is \(\frac{1}{4}\). The probability of picking a square from a box containing 3 different shapes is \(\frac{1}{3}\).

Step 1: Problem with \(\frac{1}{6}\) as answer

Consider a standard dice which has 6 faces. If asked, 'What is the probability of getting a '2' when a dice is rolled once?', the answer will be \(\frac{1}{6}\) as there is only one '2' in the dice and total outcomes are 6.

Step 2: Problem with \(\frac{1}{4}\) as answer

Imagine a bag that contains 4 differently colored balls: red, blue, green, and yellow. If asked, 'What is the probability of picking a blue ball from this bag?', the answer will be \(\frac{1}{4}\) as there is only one blue ball and total outcomes are 4.

Step 3: Problem with \(\frac{1}{3}\) as answer

Assume a box that contains 3 different shapes: square, triangle, and circle. If asked, 'What is the probability of picking a square from this box?', the answer will be \(\frac{1}{3}\) as there is only one square shape and total outcomes are 3.

Key Concepts

Probability Calculations

Understanding probability calculations is essential for solving a myriad of real-world problems, from predicting weather patterns to playing games of chance. The core of probability calculation lies in the formula: \[\begin{equation}P(Event) = \frac{Number\ of\ favorable\ outcomes}{Total\ number\ of\ possible\ outcomes}\end{equation}\]
Where 'P(Event)' represents the probability of an event occurring. In our exercise, for instance, when looking for the probability of rolling a '2' on a standard die, we observe that there is just one favorable outcome (the side showing '2') among six total possible outcomes (the six sides of the die). This leads us to a probability of \[\begin{equation}\frac{1}{6}\end{equation}\].
Probability values range from 0 (impossible event) to 1 (certain event). Fractions like \[\begin{equation}\frac{1}{6}\end{equation}\], \[\begin{equation}\frac{1}{4}\end{equation}\], and \[\begin{equation}\frac{1}{3}\end{equation}\] suggest that the events are possible but not certain, and the smaller the denominator, the higher the probability of the event occurring.

Probability with Dice

Dice games provide a classic example for learning about probability. A standard die, with six faces numbered from 1 to 6, presents a uniform probability distribution—each outcome has an equal chance of occurring when the die is fairly rolled. If we are interested in rolling a specific number, the probability is calculated by dividing the number of ways you can roll that specific number (which is always 1, unless specified otherwise) by the total number of outcomes, which for a single six-sided die, is six.

Example Problem

What is the probability of getting a '5' when a die is rolled? Since there is only one '5' on the die, and there are six possible outcomes when rolling the die, the probability is \[\begin{equation}\frac{1}{6}\end{equation}\]. This straightforward approach to probability can help demystify how odds are calculated in games of chance.

Probability with Colored Balls

A common visualization for probability problems involves selecting items, like colored balls, from a container. This scenario provides a hands-on approach for understanding probability as it imitates drawing lots or raffle tickets. For a problem with colored balls, each ball represents a different outcome. If a container holds balls of different colors in equal quantities, the probability of drawing any color is the same.

Example Problem

Assuming you have a bag with equal amounts of red, green, blue, and yellow balls, what is the probability of drawing a red ball? Since all outcomes are equally likely, and there are four balls, the probability is \[\begin{equation}\frac{1}{4}\end{equation}\]. It’s essential for students to recognize that this assumes each ball has an equal chance of being selected, irrespective of any other factors.

Probability with Shapes

Probability isn’t limited to numbers and colors; it can also extend to geometric shapes or other categorical outcomes like in our exercise scenario. When the number of different items (in this case, shapes) is small, the probability calculations remain simple and intuitive.

Example Problem

If you have a box containing a square, a triangle, and a circle, and you select one shape at random, the probability of picking out the triangle is \[\begin{equation}\frac{1}{3}\end{equation}\], assuming there is only one of each shape. Problems involving shapes often help students relate to probability through physical, tangible objects, enriching their conceptual understanding of chance and randomness.