Binomial coefficients The binomial coefficient \(\left(\begin{array}{l}n \\ k\end{array}\right)\) is an integer equal to $$ \left(\begin{array}{l} n \\ k \end{array}\right)=\frac{n !}{k !(n-k) !}=\frac{n \times(n-1) \times(n-2) \times \ldots \times(n-k+1)}{1 \times 2 \times \ldots \times k} $$ when \(k \geq 1\), or \(\left(\begin{array}{l}n \\ 0\end{array}\right)=1\) when \(k=0\). a) Using this form for the binomial coefficient, write a Python user-defined function binomial \((\mathrm{n}, \mathrm{k})\) that calculates the binomial coefficient for given \(n\) and \(k\). Make sure your function returns the answer in the form of an integer (not a float) and gives the correct value of 1 for the case where \(k=0\). b) Using your function write a program to print out the first 20 lines of "Pascal's triangle." The \(n\)th line of Pascal's triangle contains \(n+1\) numbers, which are the coefficients \(\left(\begin{array}{l}n \\\ 0\end{array}\right),\left(\begin{array}{l}n \\ 1\end{array}\right)\), and so on up to \(\left(\begin{array}{l}n \\ n\end{array}\right)\). Thus the first few lines are $$ \begin{aligned} &11 \\ &121 \\ &1331 \\ &14641 \end{aligned} $$ c) The probability that an unbiased coin, tossed \(n\) times, will come up heads \(k\) times is \(\left(\begin{array}{l}n \\ k\end{array}\right) / 2^{n}\). Write a program to calculate (a) the total probability that a coin tossed 100 times comes up heads exactly 60 times, and (b) the probability that it comes up heads 60 or more times.

Step by step solution

01

Define the binomial coefficient function

Create a Python function 'binomial' that takes two arguments, n and k. This function will calculate the binomial coefficient using the formula n! k!(n−k)! . Use Python’s math library to import the factorial function.

02

Implement the factorial function

Import the factorial function from Python’s math library to calculate fact(n), fact(k), and fact(n-k). Use these to compute the binomial coefficient.

03

Handle the special case when k=0

Modify the function to return 1 when k is 0. This is a special case as the binomial coefficient is defined as 1 when k=0.

04

Create the function to print Pascal's triangle

Write a new function to print Pascal’s triangle. This function should use the 'binomial' function to calculate each coefficient in the triangle using nested loops over the range of rows and columns.

05

Calculate the probability for heads

Write a program to calculate the total probability that a coin tossed 100 times comes up heads exactly 60 times. This involves computing 60! 60!40! 2^{−100}. Use the binomial function and probability formula to get the precise value.

06

Calculate the cumulative probability

Write another program to calculate the probability that a coin tossed 100 times will come up heads 60 or more times. This involves summing up the probabilities from 60 heads to 100 heads.

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Key Concepts

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

Probability Theory

Probability theory delves into analyzing random events and phenomena. When dealing with binomial coefficients, we often encounter problems involving probability. For example, if we repeatedly toss a fair coin, the likelihood of specific outcomes can be determined using binomial coefficients.
In probabilistic terms, the binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose k successes from n trials. Imagine tossing a coin 100 times. The probability that exactly 60 of those tosses result in heads is given by \(\binom{100}{60} / 2^{100}\).
To compute this, we follow these steps:

• First, calculate the binomial coefficient \(\binom{100}{60}\).
• Second, note that since each toss is independent with a probability of 0.5, we raise 2 to the power of the number of trials (100 in this case) to get \2^{100}\.
• Finally, divide the binomial coefficient by \2^{100}\.

The above steps allow us to understand how likely it is to achieve exactly 60 heads in 100 coin tosses. Extending this concept, the cumulative probability that we get at least 60 heads would involve summing up the probabilities from 60 heads all the way through to 100 heads.

Pascal's Triangle

Pascal's Triangle is a triangular array where each entry is a binomial coefficient. This triangle is both a mathematical curiosity and a powerful computational tool.
Pascal's Triangle has the following properties:

• Each number is the sum of the two numbers directly above it.
• The edges are all 1.
• Row n contains n+1 elements, corresponding to the coefficients \(\binom{n}{0}, \binom{n}{1}, \binom{n}{2}, \text{...} ,\binom{n}{n}\).

To generate Pascal's Triangle in Python, we can use our binomial coefficient function. For each row, calculate the binomial coefficients and display them in a format resembling a triangle.
Here's an outline of the approach:

• Iterate over the rows from 0 to the desired number of rows (say 20).
• For each row, calculate the entries using our binomial function and print them.

This simple generation showcases the elegance of binomial coefficients and their orderly arrangement in Pascal's Triangle. Additionally, Pascal's Triangle has applications in binomial expansions, combinatorial problems, and probability theory making it an invaluable concept in mathematics.

Python Programming

Python programming skills can significantly enhance our ability to solve mathematical problems. Let's focus on implementing a binomial coefficient function and using it to solve our exercise.
Start by defining the function:
import math
def binomial(n, k):
    if k == 0 or k == n:
        return 1
    return math.factorial(n) // (math.factorial(k) * math.factorial(n - k))
With this function, you can compute binomial coefficients quickly. Next, let's print the first 20 rows of Pascal's Triangle by leveraging our function:
def print_pascals_triangle(rows):
    for n in range(rows):
        row = [binomial(n, k) for k in range(n + 1)]
        print(' '.join(map(str, row)))
Finally, calculating the probability that a coin tossed 100 times comes up heads exactly 60 times:
def coin_toss_probability(n, k):
    return binomial(n, k) / (2 ** n)
print(coin_toss_probability(100, 60))
Python makes it easy to implement such mathematical functions and explore various outcomes, thereby deepening our understanding of concepts like binomial coefficients, Pascal's Triangle, and probability theory. The interplay between mathematics and programming provides a robust framework for tackling complex problems in an intuitive way.