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

When rolling two fair dice, each die has 6 possible outcomes, so there are a total of \(6 \times 6 = 36\) possible outcomes when rolling two dice.

Step 2: Define Desired Outcome Condition

We need the minimum of the two numbers to be greater than \(4\). This means both dice must show either \(5\) or \(6\).

Step 3: Identify Favorable Outcomes

For the condition to be satisfied, each die must roll a \(5\) or \(6\). The favorable outcomes are {(5,5), (5,6), (6,5), (6,6)}, totaling \(4\) outcomes.

Step 4: Calculate Probability

The probability of an event is given by \(\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\). Substituting from previous steps, \(\frac{4}{36} = \frac{1}{9}\).

Key Concepts

Favorable Outcomes

In probability, a favorable outcome is the result or set of results that satisfy the condition you are interested in. When determining favorable outcomes, you focus on the specific events that achieve the desired goal.

In the context of rolling two dice and needing the minimum number to be greater than 4, we look at those rolls where both dice show a number greater than 4. Specifically, this means both dice must roll either a 5 or a 6.

Favorable outcomes in our dice scenario are:

• (5, 5)
• (5, 6)
• (6, 5)
• (6, 6)

These all satisfy the condition of having both numbers greater than 4, totaling 4 favorable outcomes. Recognizing these helps in calculating the probability of the desired event.

Possible Outcomes

When tackling probability problems, figuring out the possible outcomes is crucial. This means identifying all the results that can occur during an experiment or event.

For two dice, each die can land on any of the 6 faces, leading to numerous combinations. To find the total possible outcomes when rolling two dice, multiply the number of outcomes of one die (6) by the number of outcomes of the other die (6). Therefore, there are \[6 \times 6 = 36\]possible outcomes.

Understanding this helps in realizing the entire space of possibilities, which is necessary to define how likely a favorable outcome is out of all these possibilities.

Dice Probability

Probability with dice revolves around understanding both favorable and possible outcomes. Once you have identified these key components, you can calculate the probability of an event occurring.

The formula for probability is:\[\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\]In our exercise, the probability of rolling two dice and having a minimum number greater than 4 is calculated using the 4 favorable outcomes (from pairs both being 5 or more) over the 36 possible outcomes:

\[\frac{4}{36} = \frac{1}{9}\]

This fraction means that for every 9 rolls of the dice, you would expect this condition to be met on average once. It highlights how probability gives us a way to predict the likelihood of events in structured settings like dice games.