Problem 15 - Chemistry The Molecular Science

Question

Suppose you flip a coin. (a) What is the probability that the coin will come up heads? (b) What is the probability that it will come up tails? (c) If you flip the coin 100 times, what is the most likely number of heads and tails you will see?

Step-by-Step Solution

Answer

(a) \( \frac{1}{2} \); (b) \( \frac{1}{2} \); (c) 50 heads and 50 tails.

Step 1: Understanding the Problem

We are dealing with a single coin flip, which is a simple two-outcome probability scenario: heads or tails. Each outcome has an equal chance of occurring when the coin is fair.

Step 2: Calculating Probability of Heads

The probability of getting heads in a single coin flip is calculated by dividing the number of favorable outcomes (heads, which is 1) by the total number of possible outcomes (heads and tails, which is 2). Thus, \( P(\text{Heads}) = \frac{1}{2} \).

Step 3: Calculating Probability of Tails

Similarly, the probability of getting tails is the number of favorable outcomes (tails, which is 1) divided by the total number of possible outcomes (2). Thus, \( P(\text{Tails}) = \frac{1}{2} \).

Step 4: Predicting Outcomes in 100 Flips

For many independent flips, such as 100, we can use the expectation to find the most likely number of heads. Since the probability of heads is \( \frac{1}{2} \), the expected number of heads is \( 100 \times \frac{1}{2} = 50 \). Similarly, the expected number of tails is the same, 50. Therefore, you are most likely to get 50 heads and 50 tails when flipping the coin 100 times.

Key Concepts

Understanding Coin Flips

A coin flip is one of the simplest examples of probability in action. When you flip a coin, there are two possible outcomes: heads or tails. Assuming the coin is fair and the environment does not interfere, the coin is equally likely to land on either side. This makes the coin flip a perfect example of an independent probability event.

Independent events are those where the outcome of one does not affect the outcome of another. So, each time you flip a coin, previous flips have no impact on the new result. This concept of independence is crucial in probability calculations involving repeated trials, such as flipping a coin multiple times.

Coin flips are a common example used to explain larger probability concepts like random variables and distribution. They are the foundation for understanding more complex scenarios where multiple independent events occur simultaneously.

Probability Calculation Basics

Probability measures how likely an event is to occur. In fundamental terms, probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For a coin flip, there are two possible outcomes: heads or tails. Both are equally likely to happen, with one favorable outcome for each.

The formula used for probability calculation is:

• Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)

In the case of a coin flip:

• Probability of Heads: There is 1 way to get heads and 2 possible outcomes overall (heads or tails), so the probability is \( P(\text{Heads}) = \frac{1}{2} \)
• Probability of Tails: Similarly, the probability of getting tails is \( P(\text{Tails}) = \frac{1}{2} \)

Each outcome being equally likely defines what is known in probability as a uniform distribution. Probability calculations are not just limited to simple events like coin tosses; they extend to more complex situations, like rolling dice, drawing cards, and even predicting weather patterns.

Calculating Expected Value in Repeated Trials

When you repeatedly perform an experiment, such as flipping a coin multiple times, the expected value helps predict the average outcome over time. The expected value is a calculated average that takes into account all possible outcomes and their probabilities.

For example, if you flip a fair coin 100 times, you can calculate the expected number of heads and tails using the formula:

• Expected Number = (Probability of Outcome) × (Total Number of Trials)

For heads in 100 flips:

• Expected Heads: \( 100 \times \frac{1}{2} = 50 \)
• Expected Tails: Similarly, the expected number of tails is also \( 50 \)

The concept of expected value is incredibly useful in predicting the number of times an event will occur in the long run. This approach can apply to more than just coin flips; it's useful in decision-making processes, finance, game theory, and various fields where probability is a key element.