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 at least one head is \( \frac{3}{4} \).

Step 1: Understand the Sample Space

When you toss two fair coins, each coin has two possible outcomes: Heads (H) or Tails (T). We must list all possible combined outcomes of the two coins. The sample space, which represents all possible outcomes, is: \( \{ (H, H), (H, T), (T, H), (T, T) \} \).

Step 2: Identify Favorable Outcomes

We want the outcomes where there is at least one head. Let's identify those outcomes from our sample space: \( (H, H) \), \( (H, T) \), and \( (T, H) \). These are the cases where there is at least one head present.

Step 3: Count Outcomes

Count the total number of possible outcomes in the sample space, which is 4 outcomes in total: \( (H, H), (H, T), (T, H), (T, T) \). Next, count the number of favorable outcomes, which is 3: \( (H, H), (H, T), (T, H) \).

Step 4: Calculate Probability

The probability of an event is the number of favorable outcomes divided by the total number of outcomes in the sample space. Therefore, the probability of having at least one head is \( \frac{3}{4} \).

Key Concepts

Sample Space

In probability, the concept of sample space is fundamental to understanding how probabilities are calculated.
A sample space includes all possible outcomes of a particular experiment or random event.
For example, when you toss two fair coins, you are creating a scenario where each coin can land either on heads (H) or tails (T).
For two coins, we need to consider the outcome of each coin simultaneously, thus forming pairs of outcomes. The complete list of these outcomes, known as the sample space, is:

• (H, H)
• (H, T)
• (T, H)
• (T, T)

Each pair represents a possible outcome from the event of tossing two coins.
Understanding the sample space is crucial as it lays the groundwork for determining probabilities, as we know the number of total possible outcomes.

Favorable Outcomes

Once we have our sample space established, the next step is to determine the favorable outcomes for the event we are interested in.
A favorable outcome is simply an outcome that matches the criteria we are looking for.
In this exercise, we want to find the probability of getting at least one head when tossing the coins.
Therefore, we look for all outcomes within our sample space that include at least one 'H'. These outcomes are:

• (H, H) - Both coins show heads
• (H, T) - The first coin shows heads
• (T, H) - The second coin shows heads

There are 3 favorable outcomes in this scenario.
Identifying these correctly is key to calculating accurate probabilities, as these are the outcomes that matter in this probability question.

Probability Calculation

Finally, with both the sample space and the favorable outcomes identified, we can proceed to calculate the probability of the event.
Probability itself is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space.
For the coin toss scenario, the sample space contains 4 outcomes:

• (H, H)
• (H, T)
• (T, H)
• (T, T)

Out of these, our favorable outcomes are 3:

• (H, H)
• (H, T)
• (T, H)

The probability of seeing at least one head in our coin tosses is thus calculated as: \[ P(\text{at least one head}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{3}{4} \]This means there is a 75% chance of getting at least one head when two coins are tossed.
This simple division formula helps to transform the complex concept of probability into something easily grasped and applied.