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Chapter 6: Problem 39

If the coin is flipped 10 times, what is the probability of exactly four or five or six heads?

Short Answer

Expert verified
The probability of getting exactly 4 or 5 or 6 heads when a coin is flipped 10 times is the sum of the probabilities calculated in Step 1, Step 2, Step 3.

Step by step solution

01

Calculation for Four Heads

Use the binomial probability formula to calculate the probability of getting exactly 4 heads. The formula is given by \( p(X=k) = C(n, k) * (p^k) * (1-p)^{(n-k)} \) where n=10 (no. of trials), k=4 (no. of successes), p=0.5 (probability of success - the coin toss is a fair event) and C(n, k) is the combination function (ways to choose k successes out of n trials).
02

Calculation for Five Heads

Repeat the same step as in Step 1, but this time change the value of k to 5 to find the probability of getting 5 heads.
03

Calculation for Six Heads

Repeat the same step as in Step 1, but this time change the value of k to 6 to find the probability of getting 6 heads.
04

Summing up the Probabilities

The final step is to add all the calculated probabilities from step 1 to step 3 to get the probability of getting exactly 4 or 5 or 6 heads when a coin is flipped 10 times.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability of Success
Understanding the concept of 'probability of success' is crucial when dealing with binomial experiments like coin flips. Simply put, it represents the likelihood of achieving a successful outcome in a single trial of an experiment.

In the context of a fair coin flip, we consider getting heads as a 'success'. Since the coin is fair, the probability of flipping heads (success) or tails (failure) is both 0.5. It is vital for students to recognize that this probability remains constant throughout each flip in a series of independent trials.
Combination Function
Another key piece of the puzzle is the 'combination function', which we denote with the symbol C(n, k). It answers the question: in how many different ways can we choose k successes out of n trials?

For instance, if we flip a coin 10 times, the combination function calculates the different ways to achieve exactly k heads. It's an integral part of the binomial probability formula because it accounts for all the possible success sequences out of the total trials.
Binomial Distribution
Now, let's dive into the 'binomial distribution'. This concept applies to a discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes/no question, and each with its own boolean-valued outcome: success/ failure.

The formula for finding the probability in a binomial distribution includes factors for the number of ways to arrange the successes, the probability of success, and the probability of failure. It's a cornerstone of discrete mathematics and is frequently used to model scenarios that match the conditions of binomial experiments like flipping a coin multiple times.
Discrete Mathematics
The field of 'discrete mathematics' encompasses various studies, including the understanding of binomial distributions and probability theory. This area of math investigates objects that can assume distinct, separated values. It's important to note that it deals with countable, often finite, elements rather than continuous systems.

Students working with problems in discrete mathematics need to exercise logical and combinatorial thinking, often using integer values. Binomial distributions, just like in our coin flip example, are a perfect illustration of discrete mathematics in action鈥攁 practical application of theoretical math concepts.

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Most popular questions from this chapter

Prove that if \(p\) ones and \(q\) zeros are placed around a circle in an arbitrary manner, where \(p, q,\) and \(k\) are positive integers satisfying \(p \geq k q,\) the arrangement must contain at least \(k\) consecutive ones.

Are the events "there are children of both sexes" and "there is at most one boy" independent?

If the coin is flipped 10 times, what is the probability of exactly five heads?

Use induction on \(n\) to prove the Binomial Theorem.

A domino is a rectangle divided into two squares with each square numbered one of \(0,1, \ldots, 6,\) repetitions allowed. How many distinct dominoes are there?
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