Problem 21 - Algebra and Trigonometry

Question

A fair coin is tossed two times in succession. The sample space of equally likely outcomes is \(\\{H H, H T, T H, T T\\} .\) Find the probability of getting $$\text{two heads.}$$

Step-by-Step Solution

Answer

The probability of getting two heads when tossing a fair coin twice is \(0.25\).

Step 1: Identifying outcomes

Firstly, we must identify all the possible outcomes of the experiment. These are \(HH\), \(HT\), \(TH\), and \(TT\). Where, \(H\) stands for a 'Head' and \(T\) stands for a 'Tail'.

Step 2: Counting favorable outcomes

Next, we count the outcomes that are 'favorable' to the event we're interested in (getting two heads). The only such outcome is \(HH\). So, there is only one favorable outcome.

Step 3: Calculating probability

Finally, the probability of an event is defined as the number of favorable outcomes divided by the total number of outcomes. In this case, both are known and so the probability is \(1 / 4 = 0.25\).

Step 4: Conclusion

So, the probability of getting two heads when tossing a fair coin twice is 0.25.

Key Concepts

Sample Space

Understanding the concept of a sample space is crucial when diving into probability exercises. Simply put, the sample space is the set of all possible outcomes that can occur in a particular experiment. Imagine it as the 'universe' of all that could happen when you perform an action, like tossing coins.

In the example of tossing a fair coin twice, the sample space consists of four possible outcomes: \( HH, HT, TH, TT \). Each outcome represents a combination of heads (H) and tails (T) across the two coin flips. Embracing the idea of a sample space lays foundational knowledge for predicting the likelihood of events, such as the chance of flipping two heads in succession.

Favorable Outcomes

Favorable outcomes are specific results within the sample space that satisfy the condition of the event we're interested in. For the scenario of two coin tosses, if we want to find the probability of flipping two heads, we must identify which outcome(s) would be considered successful.

In this case, there is only one combination that counts as a favorable outcome: \( HH \). It's essential to correctly determine all the favorable outcomes to ensure an accurate probability calculation. Sometimes there might be more than one favorable outcome, so attention to detail is key in identifying all that apply to the given event.

Probability Calculation

The probability calculation is the quantitative measure of how likely an event is to occur. The basic formula for calculating the probability of an event is to divide the number of favorable outcomes by the number of possible outcomes in the sample space.

Using our coin-toss example, the event of getting two heads (\(HH\)) has only one favorable outcome. The sample space (\( HH, HT, TH, TT \) ) includes four possible outcomes. So, the probability calculation of getting two heads is \( 1/4 \), or 0.25, which translates to a 25% chance that this event will happen on any given pair of coin tosses. Remember, the accuracy of probability calculations directly hinges on the correct determination of the sample space and the number of favorable outcomes.

Independent Events

An independent event is one where the outcome of the event is not influenced by any previous events. In other words, the result is not dependent on any other occurrences and each event stands alone. For example, when you toss a coin, the outcome of one toss has no effect on the result of the next toss.

In our exercise, each coin toss is an independent event. The first coin flip does not alter the likelihood of the result of the second coin flip. This concept assures us that the probabilities remain consistent across each flip. The independence of events in probability helps simplify our calculations and assumptions, making it a cornerstone principle in the study of chance and statistics.