Problem 27 - Algebra for College Students

Question

Solve each problem. See Examples \(3-5\) The outcome "heads" or "tails" is recorded on each toss of a coin. If we think of the outcome for 3 tosses as an ordered triple, then how many outcomes are there for 3 tosses of a coin?

Step-by-Step Solution

Answer

There are 8 possible outcomes for 3 tosses of a coin.

Step 1: Determine the possible outcomes for one toss

Each time a coin is tossed, there are 2 possible outcomes: 'heads' (H) or 'tails' (T).

Step 2: Find the total number of outcomes for three tosses

Since each toss is independent of the others, multiply the number of possible outcomes for each toss. Thus, the total number of outcomes is given by the product of the possibilities for each toss: \[ 2 \times 2 \times 2 \text{, which equals } 2^3. \]

Step 3: Calculate the total number of outcomes

Calculate \( 2^3 \) to find the total number of possible outcomes for 3 tosses of a coin: \[ 2^3 = 8 \]. Therefore, there are 8 possible outcomes.

Key Concepts

independent events

When we say events are independent, we mean that one event does not affect the outcome of another. For example, in a coin toss, getting heads or tails on one toss does not change the probabilities for the next toss. Each coin toss is an independent event.
In mathematical language, the probability of two independent events occurring is the product of their individual probabilities. If Event A and Event B are independent, then:\[ P(A \text{ and } B) = P(A) \times P(B). \]
In the given problem, each coin toss is independent. The outcome of one toss does not affect the next. This is why we can multiply the probabilities of each individual toss to find the total number of outcomes.

combinatorial outcomes

Combinatorics is the branch of mathematics dealing with combinations of objects. When tossing a coin three times, we analyze all possible ordered triples. Combinatorial outcomes help us count these possibilities.
For each coin toss, we have two choices: heads (H) or tails (T). In combinatorial language, we calculate the total number of outcomes using the multiplication principle. Since there are 2 outcomes per toss (H or T), for 3 tosses we have:\[ 2 \times 2 \times 2 = 2^3. \]
This technique computes the total outcomes by multiplying the number of choices at each step. For 3 tosses, this is 8. The possible outcomes (ordered triples) are:

• (H, H, H)
• (H, H, T)
• (H, T, H)
• (H, T, T)
• (T, H, H)
• (T, H, T)
• (T, T, H)
• (T, T, T)

This gives us a complete list of all possible outcomes.

basic probability

Basic probability involves calculating the likelihood of specific outcomes. When tossing a coin, each side (H or T) has an equal chance, making it a fair random event. The probability of one outcome (say heads) for a single toss is:\[ P(H) = \frac{1}{2}. \]
Similarly, the probability for tails is also:\[ P(T) = \frac{1}{2}. \]
When dealing with multiple independent events like 3 coin tosses, we calculate the probability of a specific sequence of outcomes by multiplying the probability of each individual event. For instance, the probability of getting heads in all three tosses is:\[ P(H \text{ and } H \text{ and } H) = P(H) \times P(H) \times P(H) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}. \]
This approach applies to any specific sequence of heads and tails in the 3 tosses.