Problem 16 - Precalculus with Limits

Question

In Exercises 15 - 20, find the probability for the experiment of tossing a coin three times. Use the sample space \( S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} \). The probability of getting exactly two tails

Step-by-Step Solution

Answer

The probability of getting exactly two tails when tossing a coin three times is \(\frac{3}{8}\).

Step 1: Identify the Favourable Outcomes

First, identify combinations from the sample space \( S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \) that contain exactly two tails. These combinations are: TTH, THT, and HTT.

Step 2: Compute the Total Number of Outcomes

The total number of outcomes is the total number of elements in the sample space, which is 8.

Step 3: Calculate the Probability

The probability of an event is given by the number of favourable outcomes divided by the total number of outcomes. In this case, the number of favourable outcomes is 3 (TTH, THT and HTT) and the total number of outcomes is 8. Hence, the probability of getting exactly two tails is \(\frac{3}{8}\).

Key Concepts

Understanding Sample Space

When we talk about probabilities, the sample space is a foundational concept. It represents all the possible outcomes of a random process. Think of it as a complete list of every single thing that could happen during an experiment. For instance, when a coin is tossed three times, the sample space includes all the possible combinations of heads (H) and tails (T) that could result. This particular example has a sample space of :
{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }. In a clear and visual way, you can imagine each coin toss as a fork in the road - first you turn either left (H) or right (T), and at each subsequent toss, you encounter similar forks. Each path is an outcome, and the collection of all paths is the sample space.

Identifying Favourable Outcomes

Focusing on favourable outcomes helps us narrow down the sample space to those outcomes that are relevant to the question at hand. These are the results we are 'rooting for' or interested in. For example, if we want to find the probability of getting exactly two tails in three coin tosses, we only count those outcomes where two of the three tosses have resulted in tails. These outcomes are { TTH, THT, HTT } . Hence, of the eight total possibilities in the sample space, only these three are favourable for the event we are calculating. It’s a bit like having a basket of fruit but just counting the apples if someone asked you what’s the probability of picking an apple.

Determining Coin Toss Probability

Now, let's calculate the coin toss probability. Probability is essentially a fraction with the number of favourable outcomes in the numerator and the total number of outcomes in the denominator. Applying this to a coin tossed three times, where we are interested in the probability of exactly two tails, we first identified that there are three favourable outcomes (TTH, THT, HTT) given in the problem. The denominator is the total number in the sample space, which is 8. Computing this we get ({3}/{8}) or 0.375. To further illustrate, imagine if you toss a coin three times, 8 sets of trials, around 3 times you’d expect to end up with exactly two tails and one head - that's the coin toss probability we've calculated. Understanding these steps ensures that you can tackle similar probability problems with confidence and accuracy.