Problem 42 - College Algebra with Modeling and Visualization

Question

Find the probability of rolling a die five times and obtaining a 6 on the first two rolls, a 5 on the third roll, and a \(1,2,3,\) or 4 on the last two rolls.

Step-by-Step Solution

Answer

The probability is \(\frac{1}{486}\).

Step 1: Identify the Success Criteria

We need to roll a die five times and achieve a specific sequence of results: two 6s, one 5, and then either a 1, 2, 3, or 4 on the last two rolls.

Step 2: Determine Individual Probabilities

For each roll, determine the probability of rolling the desired number:- Probability of rolling a 6 is \(\frac{1}{6}\).- Probability of rolling a 5 is \(\frac{1}{6}\).- Probability of rolling a 1, 2, 3, or 4 is \(\frac{4}{6} = \frac{2}{3}\).

Step 3: Compute Probabilities for Each Roll

Calculate the probabilities for each specific roll required:- First roll: P(6) = \(\frac{1}{6}\)- Second roll: P(6) = \(\frac{1}{6}\)- Third roll: P(5) = \(\frac{1}{6}\)- Fourth roll: P(1, 2, 3, or 4) = \(\frac{2}{3}\)- Fifth roll: P(1, 2, 3, or 4) = \(\frac{2}{3}\).

Step 4: Calculate Combined Probability

Multiply the probabilities of each step to find the combined probability:\[ \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) \]This simplifies to:\[ \frac{1}{6^3} \times \frac{4}{9} = \frac{1}{216} \times \frac{4}{9} = \frac{4}{1944} \].

Step 5: Simplify the Final Probability

Simplify the fraction \(\frac{4}{1944}\) by dividing both the numerator and the denominator by 4 to get:\[ \frac{1}{486} \].

Key Concepts

Dice Probability

When dealing with dice games, understanding probability is key to predicting the outcomes. Each face of a typical six-sided die (or dice) is equally likely to occur when rolled. This is because dice are designed to be fair, meaning each face—numbered from 1 to 6—has an equal chance of landing face up.

For a single die, the probability of rolling any specific number, such as a 6, is \(\frac{1}{6}\). This is because there is only one desired outcome (rolling a 6), and six possible outcomes in total. Knowing this helps us predict outcomes when rolling multiple dice or several rolls of the same die.

• Each face on a die is an equally probable event.
• The probability of rolling any single number on a fair 6-sided die is \(\frac{1}{6}\).
• Dice probability forms the basis for understanding more complex probability calculations.

Probability Calculation

Probability calculation involves determining the likelihood of different events occurring, based on possible outcomes. In our exercise, the probability of a specific sequence while rolling a die five times is being calculated.

- First, the probability of each desired event is calculated: - The chance of rolling a 6 on the first or second roll: \(\frac{1}{6}\). - Then a 5 on the third roll: \(\frac{1}{6}\). - Lastly, for the fourth and fifth rolls, a 1, 2, 3, or 4 is needed, which has a probability of \(\frac{2}{3}\) for each roll.
To find the probability of a series of events occurring in sequence, multiply the probability of each event. This helps compute the combined probability:\[\left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right)\]This product gives the joint probability of this particular sequence occurring when rolling the dice.

Mathematical Probability

Mathematical probability is a branch of mathematics that studies random events and the occurrence likelihood of different outcomes. It uses a systematic approach to quantify uncertainty and predict results.

To simplify the probability expression obtained from our exercise, \(\frac{4}{1944}\), we divide both the numerator and denominator by the greatest common divisor. This rational simplification yields \(\frac{1}{486}\), which represents the simplest form of the probability in question.

When calculating probabilities, it's important to simplify fractions to their lowest terms. This gives a clear and understandable probability value. Simplified probabilities are easier to compare with other probabilities, aiding in decision-making processes or strategy planning.

• Mathematical probability deals with predicting events using random variables.
• Fractions in probability should always be simplified to give the probability in its simplest form.
• Simplified probabilities make it easier to understand and compare different possible outcomes.