Problem 28 - Algebra and Trigonometry

Question

A coin is weighted so that tails is twice as likely as heads to occur. What probability should be assigned to heads? to tails?

Step-by-Step Solution

Answer

The probability of heads is \( \frac{1}{3} \) and the probability of tails is \( \frac{2}{3} \).

Step 1: Define the probabilities

Let the probability of heads be denoted as \( P(H) \) and the probability of tails be denoted as \( P(T) \). According to the problem, tails are twice as likely to occur as heads.

Step 2: Express the relationship mathematically

We know that \( P(T) = 2 \cdot P(H) \), since tails are twice as likely as heads to occur.

Step 3: Use the fact that total probability is 1

The sum of the probabilities for all outcomes of a coin flip must be equal to 1. Therefore, \[ P(H) + P(T) = 1 \].

Step 4: Substitute the relationship into the total probability equation

Substitute \( P(T) \) from Step 2 into the equation from Step 3: \[ P(H) + 2 \cdot P(H) = 1 \].

Step 5: Solve for the probability of heads

Combine the terms: \[ 3 \cdot P(H) = 1 \]. Divide both sides by 3 to get: \[ P(H) = \frac{1}{3} \].

Step 6: Solve for the probability of tails

Using the relationship \( P(T) = 2 \cdot P(H) \), substitute \( P(H) = \frac{1}{3} \) to get: \[ P(T) = 2 \cdot \frac{1}{3} = \frac{2}{3} \].

Key Concepts

weighted coin

When we talk about a weighted coin, we mean that the two possible outcomes—heads and tails—do not have equal probabilities. Unlike a fair coin, where heads and tails each have a probability of 0.5, a weighted coin is designed or manipulated so that one outcome is more likely to occur than the other.
In the provided exercise, tails are twice as likely to occur as heads. So, understanding this weighted nature is the first step. When you encounter a weighted coin problem, always start by identifying the given probability relationships.

probability of outcomes

The probability of an outcome is a measure of how likely that outcome is to occur. Probabilities are always between 0 and 1, where 0 means the outcome is impossible, and 1 means the outcome is certain.
For a weighted coin, let's denote the probability of heads as \(P(H)\) and the probability of tails as \(P(T)\). According to the problem, tails are twice as likely as heads, so:

• \(P(H)\) = the probability of heads
• \(P(T)\) = the probability of tails

Relationship: \(P(T) = 2 \times P(H)\). Identifying these probabilities clarifies the weighted nature of the coin.

relationship between probabilities

Probabilities in a scenario like this one are interconnected. Understanding how the probabilities of heads and tails relate to each other is pivotal.
Given the relationship \(P(T) = 2 \times P(H)\), this means every occurrence of heads, there are two occurrences of tails. Next, we use the fact that the total probability of all possible outcomes must equal 1. This ensures our probabilities are correctly normalized. Mathematically, this can be expressed as:

• \(P(H) + P(T) = 1\)

By substituting \(P(T) = 2 \times P(H)\) into this equation, we can solve for \(P(H)\) and subsequently \(P(T)\).

total probability rule

The total probability rule states that the sum of probabilities of all possible outcomes of an experiment must be 1. This is crucial for problems involving weighted probabilities.
For our weighted coin, we've identified:

• \(P(H) + P(T) = 1\)

Substituting \(P(T) = 2 \times P(H)\) into the total probability equation, we get:

• \(P(H) + 2 \times P(H) = 1\)
• \(3 \times P(H) = 1\)

From this, solving for \(P(H)\) gives \(P(H) = \frac{1}{3}\). Knowing \(P(H)\), we can then find \(P(T)\) as \(P(T) = 2 \times \frac{1}{3} = \frac{2}{3}\).