Problem 21 - Calculus for Biology and Medicine

Question

Toss two fair coins and find the probability of at least one head.

Step-by-Step Solution

Answer

The probability of getting at least one head is 0.75.

Step 1: Understanding the problem

We have two fair coins, each with two possible outcomes: Heads (H) or Tails (T). We need to find the probability of getting at least one head in one toss of both coins.

Step 2: Determine the sample space

Since each coin has two sides, the sample space is the set of all possible outcomes for the two coin tosses. These outcomes are: (H,H), (H,T), (T,H), and (T,T). This gives us a total of 4 equally likely outcomes.

Step 3: Identify favorable outcomes

We need to find outcomes where at least one coin shows heads. The favorable outcomes in our sample space that satisfy this condition are: (H,H), (H,T), and (T,H).

Step 4: Count the favorable outcomes

Counting the favorable outcomes from the sample space, we have three pairs: (H,H), (H,T), and (T,H). The number of favorable outcomes is 3.

Step 5: Calculate the probability

The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. Thus, the probability is \( \frac{3}{4} \) or 0.75.

Key Concepts

Understanding Sample Space

The sample space is a fundamental concept in probability theory. It represents the set of all possible outcomes of a random experiment. In the case of a two-coin toss, each coin can land in one of two possible ways: Heads (H) or Tails (T). Since we are tossing two coins, we need to consider all combinations of these outcomes.

By listing them, we arrive at the sample space for this experiment:

• (H, H) - both coins show heads
• (H, T) - first coin is heads, second is tails
• (T, H) - first coin is tails, second is heads
• (T, T) - both coins show tails

Thus, the sample space consists of four equally likely outcomes. Understanding and constructing the sample space is crucial because it helps in identifying all possible scenarios that can occur when performing an experiment.

Favorable Outcomes Explained

Favorable outcomes are the specific outcomes within the sample space that meet the conditions of the event we are interested in. In our exercise, we are tasked with finding the probability of getting at least one head in two coin tosses.
To identify favorable outcomes:

• Look for outcomes where at least one coin shows a head.
• In our sample space, these are (H, H), (H, T), and (T, H).

There are three favorable outcomes in total for this problem. Counting these is essential, as it provides the numerator in the probability calculation: the number of outcomes that satisfy the event divided by the total number of outcomes in the sample space.

Analyzing a Coin Toss

The simple coin toss is a classic example of a random experiment used in probability studies. It offers a perfect introduction to concepts like sample space and favorable outcomes. Each coin toss is independent, meaning the result of one coin does not affect the other.
For a single coin toss:

• There are two possible outcomes: Heads (H) or Tails (T).
• The probability of each outcome is 50% or 0.5.

When tossing two coins, combinations of results increase but each possible result remains equally likely. Understanding how multiple coin tosses create combinations helps solidify the foundational concepts of probability used to solve more complex problems. These fundamentals of coin toss lend clarity on why identifying possible and favorable outcomes is key to calculating probabilities accurately.