Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

If three coins are flipped, find the probability that exactly two heads turn up.

Step-by-Step Solution

Answer

The probability of getting exactly two heads is \(\frac{3}{8}\).

Step 1: Understand the Problem

We need to calculate the probability of getting exactly two heads when three coins are flipped. First, we determine the total number of possible outcomes.

Step 2: Determine the Total Possible Outcomes

Each coin has 2 possible outcomes: heads (H) or tails (T). Since there are three coins, the total number of possible outcomes is given by multiplying the outcomes together: \[2 \times 2 \times 2 = 8\] So, there are 8 possible outcomes when flipping three coins.

Step 3: Identify Favorable Outcomes

Next, identify the outcomes that result in exactly two heads. The combinations are: HHT, HTH, and THH. These are the sequences where exactly two out of the three coins show heads.

Step 4: Count the Favorable Outcomes

Count the outcomes identified in the previous step. We have three favorable outcomes: HHT, HTH, THH.

Step 5: Calculate the Probability

The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes. Thus, the probability is:\[P(\text{exactly 2 heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{3}{8}\]

Step 6: Simplify the Result if Necessary

In this case, the fraction \(\frac{3}{8}\) is already in its simplest form.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting and arranging objects. It's especially useful when determining the likelihood of different outcomes in probability experiments. In our exercise, we're flipping three coins and looking at how they can land, which is a classic setting for applying combinatorics.

When dealing with multiple events (like flipping several coins), each event's choices multiply together to determine the total number of possible outcomes. In this case, each coin flip has 2 possibilities: heads or tails. If you flip three coins, you use the multiplication principle to find the total number of outcomes:

• First coin: 2 outcomes (H or T)
• Second coin: 2 outcomes (H or T)
• Third coin: 2 outcomes (H or T)

This setup results in a total of \( 2 \times 2 \times 2 = 8 \) possible outcomes when flipping three coins.

Using combinatorics allows us to systematically count all possible arrangements, which is crucial for calculating probabilities.

Event Outcomes

Event outcomes are the various results that can occur from a specific experiment or trial, like flipping a coin. In this exercise, the experiment involves flipping three coins simultaneously.

Outcomes can be organized in several ways:

• Listing all scenarios, e.g., HHH, HHT, HTH, etc.
• Selecting sequences that meet certain criteria, like exactly two heads.

To solve the problem, you must first identify the specific outcomes that satisfy the event conditions. Here, we are looking for outcomes where exactly two coins show heads. The favorable outcomes are:

• HHT
• HTH
• THH

By identifying and counting these specific outcomes, we can determine the event's probability. This approach helps us focus on relevant results, reducing complexity and ensuring accuracy.

Coin Flip

A coin flip is a simple example of a binary event—meaning it has only two possible outcomes: heads (H) or tails (T). Because of its simplicity, a coin flip is often used in probability examples to illustrate basic concepts, such as independent events and probability calculations.

When you flip multiple coins, each flip stands alone. The outcome of one flip doesn't affect another; this is known as independence. For instance, the result of flipping the first coin (whether heads or tails) does not change the outcomes of the second and third flips.

Calculating the probability of specific results from multiple coin flips involves examining possible sequences of outcomes. For "exactly two heads" in three flips, we list sequences that meet the condition and count them. In our example, this gave us three favorable outcomes out of eight total. Finally, the probability is found by dividing the number of favorable outcomes by the total outcomes:\[ P(\text{exactly 2 heads}) = \frac{3}{8} \] Understanding how independent events like coin flips work will help you grasp more complex probability scenarios.