Problem 27 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The probability of heads is \( \frac{4}{5} \) and the probability of tails is \( \frac{1}{5} \).

Step 1: Identify the Problem

Determine the probabilities assigned to heads and tails, given that heads is four times as likely as tails.

Step 2: Define Variables

Let the probability of getting tails be denoted as \( P(T) = p \). Since heads is four times as likely as tails, let the probability of getting heads be denoted as \( P(H) = 4p \).

Step 3: Set Up the Equation

The total probability of all outcomes must sum to 1. Therefore, we can write the equation: \[ P(H) + P(T) = 1 \] Substitute the probabilities from the previous step: \[ 4p + p = 1 \]

Step 4: Solve for p

Combine like terms and solve for \( p \): \[ 5p = 1 \] Divide both sides by 5: \[ p = \frac{1}{5} \]

Step 5: Calculate Probabilities

Now that we have \( p \), we can find \( P(T) \) and \( P(H) \): \[ P(T) = p = \frac{1}{5} \] \[ P(H) = 4p = 4 \times \frac{1}{5} = \frac{4}{5} \]

Key Concepts

Probability Calculation

Probability helps us understand the likelihood of an event occurring. It's usually expressed as a number between 0 and 1. A probability of 0 means the event will not happen, while a probability of 1 means the event will definitely happen.
To calculate the probability, we take the number of favorable outcomes and divide it by the total number of possible outcomes. For example, the probability of flipping a fair coin and getting heads is \[ P(H) = \frac{1}{2} \]
In our weighted coin problem, however, the probabilities are different because the coin is not fair. Instead, heads is four times more likely than tails. This makes calculating the probabilities a bit different, but the total must still sum to 1.

Weighted Outcomes

Weighted outcomes refer to situations where different results have different chances of happening. This contrasts with fair situations where each outcome is equally likely.
In the case of our weighted coin, heads being four times as likely as tails changes the distribution. We need to assign probabilities that reflect this weighing.
Firstly, we let the probability of tails be \ (P(T) = p) \. Since heads is four times as likely, we set \( P(H) = 4p \). The total probability must remain 1, leading us to the equation \[ P(H) + P(T) = 1 \]

Basic Algebra

Basic algebra involves solving equations to find unknown values. In our probability problem, we had to set up and solve an equation to find the probabilities of heads and tails.
We started with the equation \[ P(H) + P(T) = 1 \] and substituted our values for the weighted coin: \[ 4p + p = 1 \]Combining like terms, we get: \[ 5p = 1 \] Next, we solve for \( p \) by dividing both sides by 5: \[ p = \frac{1}{5} \] Finally, plugging it back in gives us the probabilities: \[ P(T) = \frac{1}{5} \] and \[ P(H) = 4 \times \frac{1}{5} = \frac{4}{5} \]