Problem 28 - Algebra for College Students

Question

Three coins are tossed. What is the probability of a) getting three heads? b) not getting three heads? c) getting at least one tail?

Step-by-Step Solution

Answer

a) \( \frac{1}{8} \) b) \( \frac{7}{8} \) c) \( \frac{7}{8} \)

Step 1: Determine the sample space

List all possible outcomes when three coins are tossed. The sample space consists of the following eight outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.

Step 2: Calculate the probability of getting three heads

There is only one outcome with three heads: HHH. The probability of getting three heads is the number of favorable outcomes divided by the total number of outcomes. So, the probability is \[P(\text{three heads}) = \frac{1}{8}\].

Step 3: Calculate the probability of not getting three heads

The probability of not getting three heads is the complement of getting three heads. Since the probability of getting three heads is \( \frac{1}{8} \), the probability of not getting three heads is \[P(\text{not three heads}) = 1 - \frac{1}{8} = \frac{7}{8}\].

Step 4: Calculate the probability of getting at least one tail

To find the probability of getting at least one tail, we first find the probability of the complement event, which is getting no tails (i.e., three heads). The probability of getting no tails is \( \frac{1}{8} \). Thus, the probability of getting at least one tail is \[P(\text{at least one tail}) = 1 - \frac{1}{8} = \frac{7}{8}\].

Key Concepts

Sample Space

When dealing with probability, the **sample space** is your starting point. It represents all possible outcomes of an experiment. For instance, when tossing three coins, each coin has two possible results: heads (H) or tails (T). Combining these, the complete sample space is:
HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.
This tells us there are 8 possible outcomes in total.
Understanding the sample space is crucial because it lays the foundation for any probability calculation. It helps you see all possible results, ensuring no outcome is missed.

• Each coin toss is an independent event.
• Each combination of results forms an outcome in the sample space.
• The sample space size is important for calculating probabilities accurately.

Favorable Outcomes

Once the sample space is identified, the next step is to determine the **favorable outcomes** for the event in question.
For example, the problem asks for the probability of getting three heads (HHH):

• There is only one favorable outcome: HHH.
• Favorable outcomes are those that match the event we’re interested in.

Favorable outcomes directly relate to the sample space and are essential for computing probabilities. Particularly, they are a subset of the sample space where the event's condition is satisfied.

Complementary Events

**Complementary events** are pairs of outcomes that cover all possible outcomes of the sample space. If one happens, the other cannot, and together they account for all possible outcomes. For instance, the complement of getting three heads (HHH) is 'not getting three heads.' This includes all other outcomes from our sample space.
Here’s a useful tip: The sum of the probabilities of an event and its complement always equals 1.
For example:

• The probability of getting three heads is \(\frac{1}{8}\).
• The probability of not getting three heads is 1 - \(\frac{1}{8}\) = \(\frac{7}{8}\).

This concept makes it easier to calculate probabilities indirectly by using the complement.

Probability Calculations

To make precise **probability calculations**, follow a straightforward method:

• Identify the sample space and the number of total outcomes.
• Define the event and count the favorable outcomes.
• Divide the number of favorable outcomes by the total number of outcomes to find the probability.

For example:
a) The probability of getting three heads:
- Only one favorable outcome: HHH.
- Probability: \(\frac{1}{8}\).

b) The probability of not getting three heads:
- Complement of getting three heads.
- Probability: 1 - \(\frac{1}{8}\) = \(\frac{7}{8}\).

c) The probability of getting at least one tail:
- Complement of getting no tails (which is HHH).
- Probability: 1 - \(\frac{1}{8}\) = \(\frac{7}{8}\).

Practicing these steps with various scenarios will help solidify your understanding and accuracy in probability calculations.