Problem 21 - College Algebra

Question

For the following exercises, four coins are tossed. Find the probability of tossing four heads or four tails.

Step-by-Step Solution

Answer

The probability of tossing four heads or four tails is \( \frac{1}{8} \).

Step 1: Calculate Total Possible Outcomes

When you toss a single coin, there are two possible outcomes: heads (H) or tails (T). For four coins, each has two outcomes, resulting in a total of \( 2^4 \) possible outcomes, as they are independent events. Thus, the total number of possible outcomes is \( 16 \).

Step 2: Identify the Favorable Outcomes

Favorable outcomes refer to getting either all heads or all tails. When all are heads, the result is HHHH, and when all are tails, the result is TTTT. This gives us 2 favorable outcomes.

Step 3: Apply the Probability Formula

Probability can be calculated with the formula: \( P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \). Plugging in the values, we have \( P = \frac{2}{16} \).

Step 4: Simplify the Fraction

Now, simplify the fraction \( \frac{2}{16} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Thus, \( P = \frac{1}{8} \).

Key Concepts

Favorable Outcomes

When dealing with probability problems, it's crucial to understand what we mean by "favorable outcomes." These are the specific results we're looking for in a probability experiment. In the context of tossing four coins, favorable outcomes are the ones we want to happen, which in this case are all coins showing the same face (either all heads or all tails).

Let's break it down:

• "All heads" outcome is represented as HHHH.
• "All tails" outcome is represented as TTTT.

Both scenarios are the only favorable outcomes if our interest is getting either four heads or four tails. No other combination of heads and tails would satisfy the condition of these outcomes, which is why there are exactly 2 favorable outcomes in this scenario.

Recognizing favorable outcomes helps us focus on the results that fit our probability question. Always start by identifying these outcomes as they are essential for calculating probability.

Total Possible Outcomes

To find the probability of an event, you first need to know the total number of possible outcomes. This is the set of all potential results that could occur in a given situation. For our four coins, each coin has two possible outcomes (heads H, or tails T).

We use these steps to identify all possible outcomes:

• Each coin toss is an independent event with 2 outcomes.
• When multiple independent events occur, the total possible outcomes multiply together.
• Thus, for 4 coins: you calculate the total as \( 2^4 \), which equals 16.

This means there are 16 different ways in which the coins can land (e.g., HHHH, HHHT, ..., TTTT).

Understanding the total possible outcomes helps to provide a complete picture of what could potentially happen, and is vital for calculating the probability of any given event.

Simplifying Fractions

After identifying both the favorable outcomes and total possible outcomes, the next step in finding probability is to simplify the fraction that represents this probability. Simplifying fractions makes understanding probabilities easier.

To simplify a fraction:

• Start with the fraction formed by placing the number of favorable outcomes over the total possible outcomes. For our problem, that's \( \frac{2}{16} \).
• Identify the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 2 and 16 is 2.
• Divide both the numerator and the denominator by this GCD. Thus, we simplify \( \frac{2}{16} \) to \( \frac{1}{8} \).

Simplifying fractions is an important skill because it gives us the probability in its most understandable form. The simplified fraction \( \frac{1}{8} \) means there is a one in eight chance of tossing four heads or four tails before any coin is flipped.