Problem 21 - Algebra for College Students

Question

Six coins are tossed. Find the probability of getting at least two heads. \(\frac{57}{64}\)

Step-by-Step Solution

Answer

The probability of getting at least two heads is \( \frac{57}{64} \).

Step 1: Understand the Problem

We need to find the probability of getting at least two heads when tossing six coins. "At least two heads" means getting two, three, four, five, or six heads.

Step 2: Calculate Total Possible Outcomes

Each coin flip has two possible outcomes: heads (H) or tails (T). For six coins, the total number of outcomes is calculated by raising 2 to the power of 6: \[ 2^6 = 64 \].

Step 3: Calculate Probability of Complement Event

The complement event of getting at least two heads is getting zero or one head. We'll calculate this first and then subtract it from 1 to find our desired probability.

Step 4: Calculate Outcomes for Zero Heads

There is only 1 way to get zero heads: TTTTTT. This is 1 outcome.

Step 5: Calculate Outcomes for One Head

To get exactly one head, choose 1 out of the 6 coins to be heads, which can be done in \( \binom{6}{1} \) ways:\[ \binom{6}{1} = 6 \].

Step 6: Calculate Total Outcomes for Zero or One Head

Add the outcomes for zero and one head: \[ 1 + 6 = 7 \].

Step 7: Calculate Probability of Zero or One Head

The probability of getting zero or one head is the number of such outcomes divided by the total number of outcomes:\[ \frac{7}{64} \].

Step 8: Calculate Probability of At Least Two Heads

Subtract the probability of getting zero or one head from 1:\[ 1 - \frac{7}{64} = \frac{57}{64} \].

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that explores the study of counting, arrangement, and combination of objects. It helps in solving problems associated with probabilities, permutations, and combinations. When dealing with probability problems, such as the coin toss scenario, combinatorics allows us to determine the number of possible outcomes efficiently.

When tossing six coins, we are dealing with combinations because the coins can land in different sequences consisting of heads (H) or tails (T). Each coin has two possible outcomes, and so, with six coins, the number of potential combinations is calculated as:

• Raising the number of outcomes of a single coin to the power of the total number of coins, which is: \[ 2^6 = 64 \]. This tells us that there are 64 possible outcomes when tossing six coins.

Using combinations, we can also determine how many ways a certain number of coins can land heads up. For example, to find how many ways there are to get exactly one head, we use the combination formula: \[ \binom{n}{k} \], where \( n \) is the total number of coins (6 in this case) and \( k \) is the number of heads we want (1 in this case). Calculating this using \[ \binom{6}{1} = 6 \], shows us there are 6 different ways to achieve exactly one head.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states under a given number of observations or trials. It's particularly useful in scenarios like the coin toss, where each trial (coin flip) has two possible outcomes: heads or tails.

For the binomial distribution to apply, the trials must be independent, which means the result of one trial should not affect the others. When tossing coins, each flip is independent of the others, perfectly fitting the binomial framework. The parameters of a binomial distribution are the number of trials (\( n \)), and the probability of success (\( p \)), which in the case of a fair coin is 0.5 for heads.

• In the context of six coin tosses, we can determine the probability of any specific number of successes (heads) using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \], where \( k \) is the desired number of successes.

This formula can be used to calculate the probability of getting specific numbers of heads. However, for our problem, we use it to determine how many combinations lead to fewer than two heads, a step that is critical in finding the complement probability.

Coin Toss

A coin toss is a simple random experiment that results in one of two outcomes: heads or tails. It is a classic example of a Bernoulli trial, which serves as a fundamental building block for understanding probability. Each coin toss is an independent event, meaning the result of one toss does not influence another. This independence makes calculating probabilities straightforward when dealing with multiple tosses.

In probability exercises involving coin tosses, it's common to be tasked with calculating the likelihood of various outcomes, such as getting a certain number of heads in a series of tosses. For the scenario of tossing six coins and finding the probability of getting at least two heads:

• We first consider what outcomes count as "at least two heads". This means any outcome where there are two, three, four, five, or six heads.
• However, it's easier to start by calculating the outcomes we don't want, which are zero or one head.

By understanding that the total probability must sum to 1, the complement rule helps us quickly find the probability of "at least two heads" by subtracting the probability of zero or one head from 1. This approach simplifies the calculation process significantly.