Problem 22 - Calculus for Biology and Medicine

Question

Toss three fair coins and find the probability of no heads.

Step-by-Step Solution

Answer

The probability of no heads is \( \frac{1}{8} \).

Step 1: Understanding the Experiment

When we toss a fair coin, there are two possible outcomes: heads (H) or tails (T). For three coins, all possible outcomes can be combinations of H and T, and they are equally likely.

Step 2: Listing All Possible Outcomes

Calculate the total number of possible outcomes when three coins are tossed. Since each coin has 2 outcomes, the total number of outcomes for three coins is \(2^3 = 8\). Here are the combinations: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.

Step 3: Identifying Desired Outcomes

We need to compute the probability of getting no heads (i.e., all tails). The only such combination is TTT.

Step 4: Calculating the Probability

The probability of an event is given by the number of favorable outcomes divided by the total number of outcomes. Here, the probability of getting no heads (TTT) is \( \frac{1}{8} \).

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that studies the counting, arrangement, and combination of objects. It provides systematic ways to count and list possible outcomes in various scenarios. In probability theory, combinatorics plays a crucial role in determining the total number of possible outcomes for experiments.

• For example, when dealing with a simple experiment like tossing a coin, you can use combinatorics to calculate all potential outcomes. Each toss of a coin results in one of two outcomes: heads or tails.
• When multiple coins are tossed, the total number of possible outcomes is found by raising the number of outcomes per coin to the power of the number of coins.

By using combinatorics, we calculated that tossing three coins leads to 8 possible combinations. This helps in evaluating probabilities by identifying and classifying all potential outcomes.

Fair Coins

A fair coin is one that has an equal chance of landing heads or tails when tossed. It's a fundamental concept in probability because it introduces the idea of equally likely outcomes. With a fair coin, each outcome has a probability of 50%, or mathematically, \( \frac{1}{2} \).

• In the exercise of tossing three fair coins, the assumption is that each coin is fair, meaning each has an independent and identical probability of landing heads or tails.
• Fairness ensures that when calculating the probability of certain outcomes, like getting all tails, the approach is straightforward as each sequence has an equal likelihood.

Fair coins are typically used in probability problems because they provide a clear and unbiased foundation for calculations.

Probability Calculation

Probability calculation involves determining the likelihood or chance of a particular event happening. It's based on the ratio of favorable outcomes to the total number of outcomes, which provides a numerical value between 0 and 1.

• In the given task of finding the probability of no heads appearing when tossing three coins, we first identified the total number of possible outcomes, which is 8.
• Then, we determined the single combination, TTT, which represents the desired outcome of no heads appearing.
• Thus, the probability is calculated as \( \frac{1}{8} \), meaning out of every 8 tries, on average, getting all tails should occur once.

This straightforward method of probability calculation is essential in assessing the outcomes of random experiments.