Problem 23 - Calculus for Biology and Medicine

Question

Toss four fair coins and find the probability of exactly two heads.

Step-by-Step Solution

Answer

The probability of getting exactly two heads is \(\frac{3}{8}\).

Step 1: Understanding the Problem

First, we need to understand what is being asked. We are tossing four fair coins and need to find the probability of getting exactly two heads in those tosses. In any coin toss, there are two possible outcomes: heads (H) or tails (T).

Step 2: Determining Total Outcomes

Each coin has two possible outcomes, so for four coins, the total number of possible outcomes is calculated by multiplying the outcomes of each coin: \(2^4 = 16\).

Step 3: Counting Favorable Outcomes

Next, we need to count how many of these outcomes will result in exactly two heads. We can use combinations to find this. The number of ways to choose 2 heads out of 4 coins is given by the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(n = 4\) and \(r = 2\). So, \(\binom{4}{2} = \frac{4!}{2!2!} = 6\).

Step 4: Calculating Probability

The probability of an event is given by the number of favorable outcomes divided by the total number of outcomes. So, the probability of getting exactly two heads is \(\frac{6}{16} = \frac{3}{8}\).

Key Concepts

Combinations

In probability theory, combinations help us figure out how many ways we can select items from a larger group, without caring about the order. When trying to decide between different outcomes, combinations are a handy tool.
For example, if you are choosing two heads from four coin flips, you use the combination formula. This formula uses factorials, which are the product of all positive integers up to a number. The combination formula is written in mathematics as:

• \( \binom{n}{r} = \frac{n!}{r!(n - r)!} \)

Here, \(n\) is the total number of items to choose from (like the number of coins), and \(r\) is how many you want to choose (like deciding on heads).
In our coin-tossing example, there are 4 coins (so \(n = 4\)) and we want to choose 2 to be heads (so \(r = 2\)). By calculating \( \binom{4}{2} \), we learn there are 6 different ways to get exactly 2 heads from 4 coin flips. Combinations are essential for calculating probabilities where the sequence doesn’t matter.

Coin Tossing

Coin tossing is one of the most basic experiments in probability theory, as each toss of a coin generates two possible and equally likely outcomes: heads (H) or tails (T). Whenever you toss a coin, these outcomes are considered independent events. This means each toss does not influence another.
For multiple coin tosses, like tossing four coins, each coin contributes two outcomes, leading to exponential growth in the total outcomes. Using powers of 2, you calculate it as:

• \(2^n\)

where \(n\) is the number of coins. Therefore, with 4 coins tossed, you have \(2^4 = 16\) possible outcomes. This growing number of outcomes explains the complex nature of higher coin toss events compared to a single toss, while still maintaining simplicity in basic understanding.

Probability Calculation

Calculating probability allows us to understand how likely an event is to occur. It is computed as the number of favorable outcomes divided by the number of total possible outcomes. This forms the probability formula:

• Probability = \( \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \)

In our example of tossing four coins, we are interested in getting exactly 2 heads. We've used combinations to determine there are 6 favorable combinations. Also, previously calculated, there are 16 total possible outcomes when tossing four coins.
Using the formula:

• Probability = \( \frac{6}{16} = \frac{3}{8} \)

This means the probability of tossing exactly two heads in four coin flips is \( \frac{3}{8} \). Understanding this calculation helps in grasping more complex probability scenarios but follows the same basic toolset and logical steps.