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: Understand the Experiment

We are tossing three fair coins, and we need to find the probability of getting no heads, which means getting tails on all coins.

Step 2: Determine Possible Outcomes

Each coin has two possible outcomes: heads (H) or tails (T). For three coins, the total number of possible outcomes is calculated as: \[ 2^3 = 8 \] The possible outcomes are: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.

Step 3: Identify Favorable Outcomes

A favorable outcome is where all coins show tails, which corresponds to the sequence TTT.

Step 4: Calculate the Probability

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. Here, there is 1 favorable outcome (TTT) out of 8 total outcomes.\[ \text{Probability of no heads} = \frac{1}{8} \]

Key Concepts

Coin Toss Experiment

The coin toss experiment is a classic example of a simple probability scenario. When you toss a coin, there are two potential results: heads (H) or tails (T). In this exercise, we're exploring the outcomes for tossing three coins at the same time.

Each coin toss is independent, meaning the result of one toss doesn't affect the others. This independence is a crucial concept because it allows us to calculate the overall number of possible results by multiplying the results of each individual toss. For this three-coin toss, we multiply the possibilities for one coin (2) by itself three times (for each coin), resulting in a total of \( 2^3 = 8 \) possible combinations.

These combinations include sequences like HHH, where all coins show heads, or THT, where the first and third coin are tails, and so forth, until we get 8 unique sequences.

Favorable Outcomes

After understanding the total possible outcomes, identifying the favorable ones is the next step. A favorable outcome is simply an outcome that meets the criteria you are investigating. In our case, we are looking for the sequence of no heads, which translates to all coins showing tails.

In the list of possible sequences (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT), only one outcome matches our criterion: TTT. This is the sequence where all the coins result in tails.

By focusing on TTT as the favorable outcome, we see how crucial it is to carefully consider the criteria defined for success in your experiment. This understanding is central to performing any probability calculation effectively.

Probability Calculation

Calculating probability involves a simple formula that uses the number of favorable outcomes and the total number of possible outcomes. The formula is:

• Probability = \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)

For our exercise, the probability of all three coins showing tails (TTT) is the number of times TTT occurs divided by all possible outcomes, or sequences.

With 1 favorable outcome (TTT) and a total of 8 possible outcomes, the probability is:

• \[ \text{Probability of TTT} = \frac{1}{8} \]

This fraction tells us there is a 1 in 8 chance of all the coins showing tails in this three-coin toss experiment. Understanding this basic probability calculation method will enable you to tackle more complex probability problems with confidence.