Problem 19 - College Algebra

Question

For the following exercises, four coins are tossed. Find the probability of tossing exactly two heads.

Step-by-Step Solution

Answer

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

Step 1: Determine Total Outcomes

When four coins are tossed, each coin has two possible outcomes: heads or tails. Therefore, the total number of possible outcomes for tossing four coins is calculated by multiplying the possibilities for each coin together: \[ 2^4 = 16 \] So, there are 16 possible outcomes when tossing four coins.

Step 2: Identify the Favorable Outcomes

To find the probability of tossing exactly two heads, we must count the number of outcomes where exactly two coins show heads and the other two show tails. The combinations of two heads can be represented by: - HHTT - HTHT - HTTH - THHT - THTH - TTHH Thus, there are 6 favorable outcomes.

Step 3: Calculate the Probability

The probability of an event is given by the number of favorable outcomes divided by the total number of outcomes. Thus, the probability of tossing exactly two heads is calculated as:\[ \frac{6}{16} = \frac{3}{8} \]So, the probability of getting exactly two heads is \(\frac{3}{8}\).

Key Concepts

Combinatorics

Combinatorics is an area of mathematics focused on counting, arranging, and finding patterns in sets. It's essential when determining the number of possible outcomes in probability problems. In the context of the coin toss problem, combinatorics helps us calculate how many different ways we can arrange a certain number of heads among the tosses.
You can think of each coin toss as having two outcomes: heads (H) or tails (T). For four coins, we use combinatorics to determine all possible configurations with exactly two heads. This involves a simple combination calculation: we want to choose 2 heads out of 4 tosses.
We can express this using the binomial coefficient, commonly referred to as "n choose k," which is used to find combinations:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \(n\) is total tosses and \(k\) number of heads we want. Hence, for two heads out of four tosses, it's \(\binom{4}{2} = 6\), which confirms the 6 combinations found: HHTT, HTHT, HTTH, THHT, THTH, and TTHH.

Binomial Probability

Binomial probability involves scenarios where there are two possible outcomes for each trial, such as a coin toss (head or tail). This concept is particularly relevant when you're asked to calculate the probability of a specific number of successes, like getting exactly two heads in four coin tosses.
The binomial probability formula is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where:

• \(n\) is the number of trials (coin tosses)
• \(k\) is the specific number of successful outcomes you're interested in (two heads)
• \(p\) is the probability of success on a single trial (0.5 for heads)

In our example, letting \(p = 0.5\) for getting heads, the probability of exactly two heads in four tosses is calculated as:\[ P(X = 2) = \binom{4}{2} (0.5)^2 (0.5)^2 = \frac{6}{16} = \frac{3}{8} \]This aligns with the probability derived in the step-by-step solution.

Probability Theory

Probability theory is the mathematical framework for quantifying how likely events are to occur. It forms the basis for understanding uncertain processes, like flipping coins. Probability values always range between 0 (impossible event) and 1 (certain event).
In our exercise, the probability of tossing exactly two heads (an event) among four coins is what we're interested in calculating.
Key aspects of probability theory include:

• Total Outcomes: The complete set of potential results. Here, 16 outcomes when tossing four coins, derived from \(2^4\).
• Favorable Outcomes: The outcomes that satisfy our event's condition. Here, it's the 6 outcomes resulting in 2 heads.

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes, resulting in \( \frac{6}{16} = \frac{3}{8} \). This ratio quantifies the likelihood of observing two heads among four tosses, demonstrating a practical application of probability theory.