Problem 21 - College Algebra with Corequisite Support

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 is \( \frac{1}{8} \).

Step 1: Identify the Event Probability Space

When four coins are tossed, each coin has two possible outcomes: heads (H) or tails (T). Thus, the total number of possible outcomes for four coins is calculated as \(2^4 = 16\). These outcomes represent all the possible ways the coins can land.

Step 2: Determine Favorable Outcomes

The two desired outcomes are getting either four heads (HHHH) or four tails (TTTT). These represent the outcomes where all coins match in result as either all heads or all tails.

Step 3: Calculate Probability

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. In this case, the number of favorable outcomes (two: HHHH and TTTT) over the total outcomes (16). Thus, the probability is given by \( \frac{2}{16} \).

Step 4: Simplify the Probability

Simplify the fraction \( \frac{2}{16} \) to find the simplest form. The greatest common divisor of 2 and 16 is 2, which simplifies the fraction to \( \frac{1}{8} \). Thus, the probability of tossing either four heads or four tails is \( \frac{1}{8} \).

Key Concepts

Coin Toss Experiment

Imagine you have four coins to toss into the air. This is a classic example in probability theory known as the coin toss experiment. Each coin is independent of the others, meaning how one coin lands does not affect the others.
Each coin has two possible results:

• Heads (H)
• Tails (T)

When you toss four coins, there are several combinations possible. The total number of possible outcomes for four coin tosses is calculated using the formula:
\[2^4 = 16\]Each toss has two possibilities, which is why 2 is raised to the power of 4 (the number of coins). These 16 outcomes cover every possible combination the results can take.

Favorable Outcomes

In probability, favorable outcomes mean the specific results we're interested in for a particular question or scenario. In our coin toss problem, we're specifically looking for the outcome of either four heads (HHHH) or four tails (TTTT).
Out of the 16 total possible outcomes from tossing four coins, only two outcomes fulfill our criteria:

• HHHH
• TTTT

These are the scenarios where all the coins show the same face, either all heads or all tails. The favorable outcomes are the ones that will help us calculate the probability of the event we want to study.

Simplifying Fractions

Simplifying fractions is an essential mathematical skill, especially in probability to express the probability in the simplest form.
In this exercise, after identifying that there are 2 favorable outcomes and 16 possible outcomes, the probability fraction is:
\[\frac{2}{16}\]To simplify this fraction, we find the greatest common divisor (GCD) of the numerator and the denominator. Here, the number 2 is the GCD of both 2 and 16.
We divide both the numerator and the denominator by their GCD:

• Numerator: 2 divided by 2 equals 1
• Denominator: 16 divided by 2 equals 8

Thus, the simplified fraction is:\[\frac{1}{8}\]This simplified fraction tells us that the probability of either all heads or all tails is one in eight, written succinctly as \(\frac{1}{8}\).