Problem 18 - College Algebra

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: two heads.

Step-by-Step Solution

Answer

The probability of getting two heads when tossing a fair coin twice is \(\frac{1}{4}\).

Step 1: Determine the total number of outcomes

When a fair coin is tossed twice, there are a total of 4 outcomes {H H, H T, T H, T T}.

Step 2: Determine the favourable outcomes

The favorable outcome for getting two heads when tossing a coin twice is just one, which is {H H}.

Step 3: Calculate the Probability

The probability \(P\) of an event is the number of favourable outcomes divided by the total number of outcomes. So, the probability of getting two heads will be \(P = \frac{1}{4}\).

Key Concepts

Sample Space

When dealing with probability, understanding the sample space is crucial. It is the set of all possible outcomes that can occur from a particular experiment. For instance, when tossing a fair coin twice, the sample space consists of all the combinations of heads (H) and tails (T) that could result. These combinations are

• HH (both tosses result in heads),
• HT (first toss is a head, second is a tail),
• TH (first toss is a tail, second is a head),
• TT (both are tails).

In this case, the sample space can be written as \(\{HH, HT, TH, TT\}\). Every possible outcome is unique and equally likely if the coin is fair. Understanding the sample space is the first step to determining the probability of a specific event occurring during the experiment.

A common error students make is omitting possible outcomes from the sample space or adding duplicates. Each outcome should occur only once and all possible outcomes should be included to successfully use the sample space for probability calculations.

Favourable Outcomes

Favourable outcomes are the specific results of an experiment that we are interested in. They are a subset of the sample space and directly influence the probability of an event. In the given problem of flipping a fair coin twice, the only favourable outcome for obtaining two heads is \(\{HH\}\).

It’s important to correctly identify which outcomes are favourable based on the event we're focused on. For multiple events, this may involve listing a number of different outcomes from the sample space. But for the event 'getting two heads,' there is only one favourable outcome. Keeping a clear distinction between favourable outcomes and the rest of the sample space is key when calculating probabilities. Students sometimes confuse or miscount the favourable outcomes. Always make sure that the outcomes counted are exactly those that match the event description.

Probability Calculation

The probability calculation is the process of quantifying the chance of an event occurring. To calculate the probability, you can use the formula
\(P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes in the sample space}}\).

In our example, the event E is flipping two heads, with exactly one favourable outcome and four possible outcomes in the sample space. Therefore, the probability of the event E (two heads) happening is \(P(E) = \frac{1}{4}\) or 0.25, meaning there is a 25% chance of flipping two heads in succession. This simple division is a powerful tool in determining the likelihood of any single outcome. When approaching probability problems, it's vital to count both the favourable outcomes and the total outcomes accurately to achieve the correct calculation. Misinterpretation or miscalculation at this stage could lead to incorrect conclusions about the likelihood of an event.