Problem 28 - Calculus for Biology and Medicine

Question

Roll two fair dice and find the probability that the minimum of the two numbers will be greater than 4.

Step-by-Step Solution

Answer

The probability is \( \frac{1}{9} \).

Step 1: Determine Possible Outcomes

Firstly, when rolling two fair six-sided dice, each die has 6 faces. Thus, the total number of possible outcomes is calculated by multiplying the number of outcomes for each die: \[ 6 \times 6 = 36 \] Thus, there are 36 possible outcomes when rolling two dice.

Step 2: Identify Outcomes Where Minimum is Greater than 4

The minimum of the two numbers will be greater than 4 if both numbers rolled are either 5 or 6. Form the pairs: - (5,5), (5,6) - (6,5), (6,6) These are the possible pairs where the minimum value is greater than 4, resulting in a total of 4 successful outcomes.

Step 3: Calculate the Probability

The probability is calculated as the number of successful outcomes divided by the total number of possible outcomes. So, the probability is:\[ \frac{4}{36} = \frac{1}{9} \] This fraction represents the likelihood that the minimum value rolled is greater than 4.

Key Concepts

Dice

Rolling dice is a common and classic example of exploring probability. When we talk about dice in probability, we're usually referring to standard six-sided dice. Each side of a die is marked with dots ranging from 1 to 6. So, when you roll one die, the outcomes can be any of these six numbers: 1, 2, 3, 4, 5, or 6.

In probability exercises, rolling dice helps us visualize simple random experiments. They are fair, meaning that each number from 1 to 6 has an equal chance of appearing. Dice can be used individually, or as in our exercise, multiple dice can be used together to explore more complex scenarios.

• Rolling one die: You get 6 possible outcomes.
• Rolling two dice: You calculate possible outcomes by multiplying the outcomes of each, i.e., \(6 \times 6 = 36\).

Understanding dice rolls provides foundational insights into probability, which in turn builds more complex problem-solving skills.

Outcomes

The concept of outcomes is essential in understanding probability. An outcome is the result of a single trial of a probabilistic experiment. When dealing with dice, each roll or each combination of rolls can represent a unique outcome.

In our example with two dice, each roll of the dice results in a pair of numbers. These pairs are distinct outcomes within the context of our experiment. With two six-sided dice, there are \(6 \times 6 = 36\) possible outcomes because each die can land on any of its 6 faces independently.

• Single die outcomes: 1 through 6
• Two dice outcomes: Pairs like (1,1), (1,2), ..., through to (6,6)

Focusing on the outcomes allows us to identify specific results that meet the conditions of a problem. For instance, if we want the minimum of the two numbers to be greater than 4, we identify the specific outcomes, like (5,5) and (5,6), that satisfy this condition.

Fractions

Fractions play a vital role in expressing probabilities. They provide a way to convey the likelihood of a specific outcome occurring within the context of all possible outcomes. When calculating probability, the number of favorable outcomes is divided by the total possible outcomes, resulting in a fraction.

In our dice problem, there are 4 favorable outcomes for the condition that the minimum number is greater than 4, and a total of 36 possible outcomes when rolling two dice. The probability is calculated as: \[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{4}{36} = \frac{1}{9} \]

• Favorable outcomes: Situations that meet your condition.
• Total outcomes: All possible results of the experiment.

Using fractions to express probabilities makes it easier to communicate risk and possibility clearly, and is a critical skill in statistical analysis and everyday decision-making.