Problem 36 - Calculus for Biology and Medicine

Question

A loaded die has probability \(0.5\) of rolling a 6 and probability \(0.1\) of rolling each of the other five numbers. Find the probability of rolling a 6 three times in a row.

Step-by-Step Solution

Answer

The probability of rolling a 6 three times in a row is 0.125.

Step 1: Understanding the Dice Probabilities

A loaded die has a probability of 0.5 for rolling a 6 and a probability of 0.1 for rolling each of the numbers 1 through 5.

Step 2: Calculating Single Event Probability

The probability of rolling a 6 in a single roll of this loaded die is 0.5.

Step 3: Applying the Multiplication Rule for Independent Events

Since each roll of the die is independent, the probability of rolling a 6 three times in a row is found by multiplying the probability of rolling a 6 for each roll: \[ P(6 \text{ on first roll}) \times P(6 \text{ on second roll}) \times P(6 \text{ on third roll}) \].

Step 4: Performing the Multiplication

Compute the probability by multiplying the probabilities: \[ 0.5 \times 0.5 \times 0.5 = 0.5^3 = 0.125 \].

Key Concepts

Understanding Loaded Dice

A loaded die is not your regular fair die where each side has an equal chance of landing face up. Instead, a loaded die is weighted or biased to favor certain outcomes over others. This means certain numbers have a higher probability of being rolled compared to what you would expect with a fair die. For example, if you have a loaded die where the probability of rolling a 6 is 0.5, this is much higher than the 1/6 (approximately 0.1667) probability in a fair die.
Loaded dice are often used in discussions of probability to illustrate how skewed systems can affect outcomes and differ significantly from fair scenarios. When working with loaded dice, it's crucial to first understand the given probabilities of each of the outcomes as they will affect any calculation of probabilities for multiple rolls.

Independent Events Explained

In probability theory, understanding independent events can significantly simplify solving complex problems. Events are said to be independent when the occurrence of one event does not affect the probability of another event occurring.
For instance, when rolling a die, the outcome of one roll does not affect the outcome of the next roll. This lack of influence between events is a key component in determining probabilities over multiple instances.

• For example, if you roll a loaded die and get a 6, the result of this roll doesn’t change the probabilities for your next rolls.
• Consequently, if you are rolling this die three times, each roll remains independent of the others.

Appreciating the independence of events is essential for accurate probability calculations, especially in cases of repeated trials like rolling the die multiple times.

Applying the Multiplication Rule

The Multiplication Rule is a vital tool in probability when dealing with independent events. It states that the probability of two or more independent events occurring in sequence is the product of their individual probabilities.
This rule is extremely useful when, for example, calculating the probability of rolling a 6 three times in a row with a loaded die. Since each roll is independent, you can multiply the probability of getting a 6 on each roll to get the total probability for the sequence.

• Given the probability of rolling a 6 on each roll is 0.5, the probability for three consecutive rolls would be calculated as: \[ P(6 ext{ on first roll}) \times P(6 ext{ on second roll}) \times P(6 ext{ on third roll}) = 0.5 \times 0.5 \times 0.5 \]
• Simplifying, you find this results in \[ 0.5^3 = 0.125 \].

By applying the multiplication rule, you can easily calculate such probabilities for any number of independent events.