Problem 24 - Calculus for Biology and Medicine

Question

Toss four fair coins and find the probability of three or more heads

Step-by-Step Solution

Answer

The probability of getting three or more heads is \( \frac{5}{16} \).

Step 1: Understanding the Problem

We need to calculate the probability of getting three or more heads when tossing four fair coins. The possible outcomes for each coin are either heads (H) or tails (T).

Step 2: Identifying Total Outcomes

Each coin flip has two possible outcomes (H or T), and since there are four independent coin tosses, the total number of possible outcomes is given by: \( 2^4 = 16 \).

Step 3: Defining Favorable Outcomes

We are interested in finding the probability of getting exactly three heads and exactly four heads. We will consider the number of ways to get each of these outcomes from the four tosses.

Step 4: Calculating Outcomes for Three Heads

To get exactly three heads in four tosses, we can choose three coins to show heads. The number of ways to choose which three coins will be heads is given by \( \binom{4}{3} \). This calculation results in \( \binom{4}{3} = 4 \) favorable outcomes.

Step 5: Calculating Outcomes for Four Heads

If all four coins show heads, there is only one possible outcome: HHHH. Therefore, there is 1 favorable outcome for getting four heads.

Step 6: Adding Up Favorable Outcomes

The total number of favorable outcomes for getting three or more heads is the sum of the outcomes for three heads and four heads. Thus, there are \( 4 + 1 = 5 \) favorable outcomes.

Step 7: Finding the Probability

The probability of getting three or more heads is the number of favorable outcomes divided by the total number of outcomes. Therefore, the probability is given by \( \frac{5}{16} \).

Key Concepts

Coin Toss

A coin toss is a simple experiment with only two possible outcomes: heads (H) or tails (T). In a fair coin, the likelihood of either outcome is equal, meaning each has a probability of 0.5. Tossing a coin multiple times is a common method to demonstrate concepts of probability and randomness. When you toss a coin, it's an independent event. This means each toss doesn't affect the others. It's important to understand independence when assessing the outcomes of multiple coin tosses. For example, when tossing four coins, each flip is an independent event. This setup allows us to use various probability rules to solve more complex problems. By examining multiple data points (like the number of heads in multiple tosses), you explore deeper probabilistic behavior.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, arrangements, and counting. It plays a crucial role in probability because it helps us understand how to count possible outcomes. In our task, we aim to find how many ways we can get a certain number of heads when tossing four coins. To do this, we use combinations, denoted as \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, it's used to find the number of ways to get exactly three heads out of four coins. Using the formula: \[ \binom{4}{3} = 4 \] This tells us there are 4 ways to get 3 heads out of 4 tosses. Understanding combinatorics is essential to solve probability problems since it simplifies the process of calculating probable and possible outcomes without listing all possibilities.

Binomial Distribution

The binomial distribution is a statistical distribution that represents the probability of a given number of successes in a set of independent experiments. Each experiment is called a trial, and in our scenario, each coin toss is a trial. A success could be defined as getting heads on the coin flip. The binomial distribution is characterized by having a fixed number of trials, two possible outcomes (success or failure), a constant probability of success in each trial, and independence across trials.We calculate a specific probability using the binomial distribution by the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where:

• \( P(X = k) \) is the probability of getting k successes (e.g., heads).
• \( n \) is the total number of trials (coin tosses).
• \( k \) is the number of successes (number of heads).
• \( p \) is the probability of success on an individual trial (0.5 for a fair coin).

Applying this to our example helps find the probability of 3 or 4 heads, demonstrating both the essence and utility of the binomial distribution in probability theory.