Question

You flip a coin three times. (a) What is the probability of getting heads on only one of your flips? (b) What is the probability of getting heads on at least one flip?

Short answer

(a) \( \frac{3}{8} \); (b) \( \frac{7}{8} \).

Step-by-step solution

Understand the Problem

We need to determine the probability of certain outcomes when flipping a fair coin three times. We will first calculate the probability of getting exactly one head (part a) and then the probability of getting at least one head (part b).

List Possible Outcomes

When you flip a coin three times, there are 2 possible outcomes for each flip (heads or tails). Thus, the total number of outcomes is given by \( 2^3 = 8 \). These outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Calculate Probability for Part (a)

We need to find the number of outcomes where there is exactly one head. The outcomes are: HTT, THT, TTH. There are 3 outcomes with exactly one head. Thus, the probability is \( \frac{3}{8} \).

Calculate Probability for Part (b)

For at least one head, we find the complement (no heads at all). The outcome with no heads is TTT, which has a probability of \( \frac{1}{8} \). Therefore, the probability of at least one head is \( 1 - \frac{1}{8} = \frac{7}{8} \).

Key concepts

Coin Flipping

Coin flipping is a classic example in probability theory often used to illustrate basic concepts. When you flip a coin, it has two sides: heads and tails. Each flip is an independent event, meaning the result of one flip doesn't affect the next. This independence is important because it allows us to calculate probabilities assuming each coin flip has an equal chance of landing heads or tails, typically a probability of 0.5 or 50% for each result.
In exercises like the one presented, you're often asked to evaluate the likelihood of various sequences over multiple flips. Even with a simple tool like a coin, understanding the outcomes and their probabilities helps set the foundation for more complex probability problems.

Outcome Calculation

When dealing with probability in coin flipping, calculating the number of possible outcomes is a key step. With a single coin flip, the outcomes are straightforward: heads or tails.
But when flipping the coin multiple times, say three as in the exercise, the number of outcomes increases exponentially according to the formula \(2^n\), where \(n\) is the number of flips.
For three flips, you get \(2^3 = 8\) different sequences, such as HHH, HHT, HTT, and so on. Listing all possible sequences is useful to visualize and calculate probabilities for specific outcomes, such as getting one head or at least one head in a given series of flips.

Complementary Probability

Complementary probability is a useful concept in calculating the likelihood of an event by considering the event's opposite. In probability, the complement of an event \(A\) is usually denoted as \(\bar{A}\), and its probability is calculated as \(1 - P(A)\).
This concept simplifies problems like part (b) of our exercise. Instead of counting all sequences with at least one head, you find it easier to calculate the probability of getting no heads (TTT) and subtract from 1. In this exercise, there's a single sequence TTT with no heads, offering a probability of \(\frac{1}{8}\). Hence, getting at least one head is \(1 - \frac{1}{8} = \frac{7}{8}\).
Complementary probability is a time-saving strategy, especially when directly counting outcomes can be cumbersome.

Discrete Probability

Discrete probability deals with events that occur in countable sample spaces, such as the flip of a coin. In such cases, it's useful for problems where outcomes are separate and distinct. Each outcome here, such as HTT or TTH, is separate and can be counted.
When you flip a coin, you get a finite number of possible outcomes, eight in the case of three flips. Discrete events resolve into clear results that can be directly enumerated and thereby make probability calculations straightforward.
Calculating probabilities in discrete spaces involves counting favorable outcomes and dividing by the total number of possible outcomes. For instance, for part (a) in the exercise—obtaining exactly one head—the favorable outcomes (HTT, THT, TTH) are countable (3 in total), yielding a probability of \(\frac{3}{8}\). Understanding discrete probability helps in dealing with events that may look complex but resolve into simple counts of outcomes.