Problem 9 - Algebra for College Students

Question

Four coins are tossed. Find the proba- bility of tossing each of the following events: Four heads \(\frac{1}{16}\)

Step-by-Step Solution

Answer

The probability of tossing four heads is \(\frac{1}{16}\).

Step 1: Calculate Total Outcomes

Each coin has 2 possible outcomes: heads (H) and tails (T). When four coins are tossed, the total number of outcomes is computed by raising the possibilities of one coin to the power of the number of coins. Thus, the total outcomes are \[2^4 = 16\]

Step 2: Identify Favorable Outcomes

To get four heads, all four coins must land heads. The sequence (H, H, H, H) represents this outcome. Hence, there is only 1 favorable outcome for getting four heads.

Step 3: Calculate Probability

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes. Therefore, the probability of tossing four heads is:\[\frac{1 \text{ favorable outcome}}{16 \text{ total outcomes}} = \frac{1}{16}\]

Key Concepts

Coin Tosses

Understanding the mechanics of coin tosses is essential in probability theory. A coin toss generates two possible outcomes: heads (H) or tails (T). When multiple coins are tossed, these individual outcomes combine to form a set of results.

For instance, in the exercise where four coins are tossed, each coin flip is independent, meaning the result of one flip does not affect the others. This independence is crucial because it allows us to calculate the total number of outcomes by multiplication.

For a single coin, there are 2 outcomes. For four coins, you calculate the total outcomes by raising 2 (the number of outcomes for one coin) to the power of 4 (the number of coins): \[2^4 = 16\] This shows how quickly the number of possibilities grows as you add more coins to the experiment.

Favorable Outcomes

Favorable outcomes are those specific results that meet the criteria we're interested in. In probability theory, identifying the favorable outcome is critical in determining the likelihood of an event.

In our exercise, the goal is to get four heads when tossing four coins. The favorable outcome for this specific event is a sequence where all coins show heads, written as (H, H, H, H).

It's important to note that even though there are 16 total possibilities when tossing four coins, only one of them fits our criteria. Recognizing and correctly counting favorable outcomes is a key step in calculating probabilities.

Total Outcomes

Total outcomes encompass all possible results that could occur when carrying out an experiment or event. In probability, understanding total outcomes assures us of comprehensively considering every possibility.

For the example of tossing four coins, as previously calculated, there are 16 total outcomes. This total comes from the multiplicative effect of each coin having two potential results, and it’s vital in providing the denominator for the probability fraction.

The structure of total outcomes in probability is usually recognized as the basis: a complete list of every possible occurrence. Having this full picture allows us to accurately gauge how likely specific results are, by comparison to the entire set of possibilities.