Problem 59 - A First Course in Probability

Question

Independent flips of a coin that lands on heads with probability \(p\) are made. What is the probability that the first four outcomes are (a) \(H, H, H, H ?\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H, H, H\) occurs before the pattern \(H, H, H, H ?\) Hint for part \((c):\) How can the pattern \(H, H, H, H\) occur first?

Step-by-Step Solution

Answer

In summary: (a) The probability of the first four outcomes being H, H, H, H is \(p^4\). (b) The probability of the first four outcomes being T, H, H, H is \((1-p)p^3\). (c) The probability of the pattern T, H, H, H occurring before the pattern H, H, H, H is \(1 - p^4\).

Step 1: (a) Probability of H, H, H, H Outcomes

Since the flips are independent, to find the probability of the first four outcomes being H, H, H, H, we simply multiply the probability of a single H outcome by itself four times: \[P(H, H, H, H) = p \times p \times p \times p = p^4\]

Step 2: (b) Probability of T, H, H, H Outcomes

Similarly to part (a), we can find the probability of T, H, H, H outcomes by multiplying the probabilities of individual outcomes. The probability of T occurring is (1-p), since either H or T can occur: \[P(T, H, H, H) = (1-p) \times p \times p \times p = (1-p)p^3\]

Step 3: (c) Probability of T, H, H, H Occurring before H, H, H, H

To find the probability of the pattern T, H, H, H occurring before the pattern H, H, H, H, we need to consider the following scenario: H, H, H followed by T, H, H, H occurring in sequence. In order for the H, H, H, H pattern to occur first, the first three flips must be heads, followed by another head. Thus, the probability for H, H, H, H to occur first is given by the same probability as in part (a). Now, we want to find the probability of the opposite scenario, where T, H, H, H occurs before H, H, H, H. This opposite scenario has a probability of 1 minus the probability of H, H, H, H occurring first: \[P(T, H, H, H \text{ before } H, H, H, H) = 1 - P(H, H, H, H \text{ occurs first})\] Substitute the probability we found in part (a): \[P(T, H, H, H \text{ before } H, H, H, H) = 1 - p^4\]

Key Concepts

Independent Events

When flipping a coin, each toss is an event that does not affect the outcome of the subsequent tosses. This concept is known as independent events in probability theory. To illustrate, consider flipping a coin that lands heads up with probability \( p \); each flip is independent of the last. Thus, if you are looking to calculate the probability of two independent events both occurring, you multiply their individual probabilities together. For example, the probability of obtaining heads on the first flip (\textbf{H}) and then again on the second flip is \( p \times p \), or \( p^2 \). This rule of multiplication applies generally in scenarios with independent events, which is why the solutions for parts \(a\) and \(b\) involve multiplying the probabilities of individual coin toss outcomes.

Probability Theory

The field of probability theory is concerned with predicting the likelihood of events. Be it the toss of a coin or more complex outcomes, the theory provides a mathematical framework for quantifying these chances. Probability values range from 0 (the event never occurs) to 1 (the event always occurs), and are often expressed as a fraction or percentage. Real-world situations often translate into probability problems that require a good understanding of the basic principles, such as the concept of independent events, as explained earlier. For the coin toss exercise, probability theory dictates the calculations for the likelihood of certain sequences of heads \(H\) and tails \(T\) appearing.

Combinatorics

Combinatorics plays a central role in probability theory, especially when dealing with discrete outcomes. It is the branch of mathematics concerning the counting, combination, and permutation of sets of elements. In the context of a coin toss, each flip has two possible outcomes — heads or tails. However, combinatorial principles come into play when we consider sequences of multiple flips. For instance, the number of ways to get two heads in four tosses can be worked out using combinatorial logic. While the coin flip exercise given does not require complex combinatorial calculations, understanding the basics of combinatorics helps in grasping more involved probability problems.

Binomial Outcomes

The concept of binomial outcomes is related to the occurrence of two mutually exclusive events, such as flipping a coin where only heads or tails can occur. The binomial distribution is a common probability distribution in statistics that models the number of successes in a fixed number of independent trials, each with the same probability of success. In the case of our coin flip problem, the outcomes of each sequence (\(H, H, H, H\) and \(T, H, H, H\)) can be considered binomial in nature, where 'success' is defined as getting the specific outcome we are calculating the probability for. The parts \(a\) and \(b\) demonstrate how the probabilities for binomial outcomes are obtained by multiplying the individual probabilities of each 'trial' (or flip).