Problem 1 - Algebra 2

Question

Find each probability if a coin is tossed 3 times. \(P(\text { exactly } 2 \text { heads })\)

Step-by-Step Solution

Answer

The probability is \(\frac{3}{8}\).

Step 1: Understand the Sample Space

When a fair coin is tossed 3 times, each toss has 2 possible outcomes: Heads (H) or Tails (T). This means there are a total of \(2^3 = 8\) possible outcomes. The complete sample space is: HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT.

Step 2: Identify Desired Outcomes

We are interested in finding the number of outcomes where exactly 2 heads appear. By examining the sample space, the outcomes with exactly 2 heads are: HHT, HTH, and THH.

Step 3: Calculate Probability

There are 3 outcomes with exactly 2 heads out of a total of 8 possible outcomes. Therefore, the probability of getting exactly 2 heads is \(\frac{3}{8}\).

Key Concepts

Sample Space

In probability, the sample space is a fundamental concept that encompasses all possible outcomes of a random experiment. When a coin is tossed, each flip results in either a head (H) or a tail (T). For more complex scenarios, like tossing the coin multiple times, the sample space expands.
For our exercise of tossing a coin three times, the total number of possible outcomes is calculated as the power of 2 raised to the number of tosses: \(2^3 = 8.\)This means there are eight different possible sequences that could occur.
These sequences or outcomes are represented as follows:

• HHH
• HHT
• HTH
• THH
• HTT
• THT
• TTH
• TTT

Understanding the complete sample space is crucial for determining probabilities because it provides the full set of events that could potentially occur.

Outcomes

Each sequence of results from the coin tosses represents a unique outcome. When we discuss outcomes, we focus on particular events that we are interested in.
For this exercise, the goal is to find the probability of exactly two heads appearing in three tosses.
By observing the sample space, we identify the specific outcomes that match this criteria. These outcomes are:

• HHT
• HTH
• THH

These three outcomes reflect the different combinations in which exactly two heads can occur. Outcomes are vital because they help us isolate the events that we are calculating the probability for, based on specific conditions or criteria.

Coin Toss Probability

The coin toss is a classic example of a simple probability event. When you flip a fair coin, you have two equally likely outcomes: heads or tails, each with a probability of \(\frac{1}{2}.\)In more complex scenarios involving multiple coin tosses, such as three times in this exercise, the principles still apply. Each toss is independent, meaning the result of one does not affect the others.
To find the probability of obtaining exactly two heads when tossing three coins, we use the principle of dividing the desired outcomes by all possible outcomes.
Given the complete sample space of 8 outcomes and 3 that fit the criteria of having two heads, we use the formula:\(P( ext { exactly 2 heads }) = \frac{3}{8}.\)This calculation is a direct illustration of how probability can be applied to simple repetitive scenarios like coin tosses.

Binomial Probability

Binomial probability is a powerful statistical concept used when an event happens a fixed number of times, each with two possible outcomes. This makes it perfect for situations like coin tosses.
The general formula for binomial probability is \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\), where:

• \(n\) is total number of trials (coin tosses)
• \(k\) is the number of successful outcomes (getting heads)
• \(p\) is the probability of success on an individual trial

Applying this to our example, if we want exactly 2 heads out of 3 tosses, the parameters are \(n = 3, k = 2, p = \frac{1}{2}\). The binomial coefficient \(\binom{3}{2}\) calculates the number of ways to achieve 2 successes (heads) in 3 trials, leading to:
\[P(X = 2) = \binom{3}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 = 3 \, \cdot \, \frac{1}{4} \, \cdot \, \frac{1}{2} = \frac{3}{8}\]This demonstrates how the binomial model offers a consistent method to calculate probabilities for events with fixed, independent trials.