Problem 55 - Algebra for College Students

Question

Nine coins are tossed. Find the probability of getting three heads and six tails.

Step-by-Step Solution

Answer

The probability is \( \frac{21}{128} \) or approximately 0.1641.

Step 1: Understand the Problem

We have 9 coins tossed, and we need to calculate the probability of getting exactly three heads and six tails in these tosses. Each coin flip is independent, meaning the outcome of one coin does not affect another.

Step 2: Identify the Type of Probability Problem

This is a binomial probability problem where we need to find the probability of getting exactly a certain number of successes (heads) in a fixed number of independent trials (coin tosses).

Step 3: Determine the Parameters of the Binomial Distribution

In this problem, we have:- Number of trials, \( n = 9 \) (number of coins tossed)- Number of successes, \( k = 3 \) (getting heads)- Probability of success on each trial, \( p = 0.5 \) (probability of getting a head in one coin toss)- Probability of failure, \( 1-p = 0.5 \) (probability of getting a tail).

Step 4: Calculate the Binomial Coefficient

The binomial coefficient is a combination that tells us how many ways we can choose 3 heads out of 9 tosses. It is calculated as:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Substitute \( n = 9 \) and \( k = 3 \):\[\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84\]Thus, there are 84 ways to get 3 heads and 6 tails in 9 tosses.

Step 5: Use the Binomial Probability Formula

The probability of getting exactly \( k \) heads in \( n \) tosses is given by the formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]Substitute the known values \( n = 9 \), \( k = 3 \), \( p = 0.5 \), and \( 1-p = 0.5 \):\[P(X = 3) = 84 \times (0.5)^3 \times (0.5)^6 = 84 \times (0.5)^9\]

Step 6: Compute the Probability

Now, compute \((0.5)^9\):\[(0.5)^9 = \frac{1}{2^9} = \frac{1}{512}\]Hence, the probability is:\[P(X = 3) = 84 \times \frac{1}{512} = \frac{84}{512} = \frac{21}{128}\]Therefore, the probability of getting exactly 3 heads and 6 tails is \( \frac{21}{128} \approx 0.1641 \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics concerned with analyzing random phenomena. When we talk about probability, we're essentially asking, "How likely is an event to occur?" This is pivotal in understanding various real-world scenarios, from weather forecasts to deciding the outcomes of games. In the context of tossing coins, each flip represents a random event with two possible outcomes: heads or tails.



Here, probability is calculated as a number between 0 and 1. A probability of 0 means an event cannot happen, while a probability of 1 indicates certainty. In coin tossing, assuming a fair coin, each outcome (heads or tails) has a probability of 0.5.

Binomial Distribution

The binomial distribution is a common discrete probability distribution. It describes the number of successes in a fixed number of independent trials of a binary experiment—experiments with two possible outcomes (like tossing coins).

In our exercise, we deal with a binomial distribution because we are calculating the likelihood of a certain number of successes (heads) out of 9 trials (tosses). The parameters defining a binomial distribution are:

• n (number of trials): Total number of coin tosses. In this case, it's 9.
• k (number of successes): Number of times we want a particular outcome, here 3 heads.
• p (probability of success in one trial): The chance of getting heads in a single toss, 0.5.

The binomial probability formula helps us find the probability for getting exactly k successes: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]The distribution is named because it's related to the 'binomial theorem' that involves powers of sums.

Combinatorial Mathematics

Combinatorial mathematics is the study of counting, arrangement, and combination of objects. It's a vital tool in probability theory, especially in problems involving randomness.

In finding the probability of getting 3 heads in 9 coin tosses, we use a concept from combinatorics called the "binomial coefficient," represented as \(\binom{n}{k}\). This coefficient tells us how many different ways we can choose a subset of items (e.g., select 3 heads) from a larger set of items (9 tosses).

The formula for the binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where "!" denotes a factorial, the product of all positive integers up to that number. For example, 3! = 3 × 2 × 1.

• In our coin-toss exercise, \(\binom{9}{3}\) equals 84, meaning there are 84 different ways to arrange three heads among nine tosses.

Understanding these combinations is crucial for calculating probabilities in situations where order does not matter.