Problem 25 - Calculus for Biology and Medicine

Question

Roll a fair die twice and find the probability of at least one \(4 .\)

Step-by-Step Solution

Answer

The probability of rolling at least one 4 is \(\frac{11}{36}\).

Step 1: Understand the Problem

We need to find the probability of rolling at least one 4 when a fair six-sided die is rolled twice. This involves determining the chances of getting at least one 4 in the outcomes of two dice rolls.

Step 2: Calculate Total Number of Outcomes

Since each die has 6 faces, the total number of possible outcomes when rolling the die twice is calculated as 6 (faces on the first die) multiplied by 6 (faces on the second die), which equals \(6 \times 6 = 36\).

Step 3: Calculate the Complement Probability

Instead of directly calculating the probability of rolling at least one 4, we'll find the complement - the probability of rolling no 4s at all. For each roll, the probability of not rolling a 4 is 5/6. Hence, for two rolls, the probability of not getting any 4 is \(( \frac{5}{6} )^2 = \frac{25}{36}\).

Step 4: Use Complement to Find Desired Probability

The probability of getting at least one 4 is the complement of the probability of getting zero 4s. Thus, we subtract the probability of rolling no 4s from 1: \(P(\text{{at least one 4}}) = 1 - \frac{25}{36} = \frac{11}{36}\).

Key Concepts

Complement Rule

The complement rule is a nifty trick in probability that simplifies calculations. When we talk about probabilities, sometimes it's easier to figure out the chance of something **not** happening, rather than it actually happening. Think of the complement as the opposite. 
A clear case is trying to find the probability of at least one occurrence of an event. Instead of directly calculating it, you calculate the probability of the event not happening at all and subtract from 1. For instance, if we are trying to find the probability of rolling at least one 4 in two dice rolls, calculating directly might seem tricky.
With a complement rule, we first find out how likely it is to roll **no 4s at all**. In this exercise, if each die has 6 faces, the chance of not rolling a 4 in one toss is 5/6. If we roll twice, the probability becomes rrac{25}{36}r. Then, subtracting this from 1 gives the probability of at least one roll showing a 4.

Probability of Dice Rolls

Dice rolls are classic examples when it comes to understanding probability. A fair six-sided die has numbers from 1 to 6, each with an equal chance of landing face-up. When you plan to roll a die, think of it as having 1 out of 6 shots for each number, or, in other words, its probability is 1/6. 
Rolling the die more than once adds a layer of fun to probability calculations. Every roll is independent, meaning the result of one roll does not change or affect the next. If, like in this exercise, you roll the die twice, you must consider each roll separately but also explore possible pairs.
If you're asking about the chance of at least one 4 appearing, you find probabilities for all rolls ending without a 4. This is 5/6 for each die that does not show a 4, multiplying them for an overall view when rolled twice.

Total Number of Outcomes

Understanding the total number of outcomes is crucial when dealing with probability. It’s the starting line where all probability calculations begin. For one six-sided die, you have 6 possible outcomes.
However, rolling two dice does not double but expands possibilities because each face on the first die can pair with any face on the second die. As a result, the number of outcomes is the product of the number of faces on each die. 
rFor our current problem, that's 6 faces on the first die times 6 faces on the second die, summing up to 36 possible outcomes overall.
Exploring two dice rolls gives you combinations like (1,1), (1,2), up to (6,6). It's this exhaustive list of possible pairs or combinations of face results that helps calculate the probability more precisely.