MHB Prove Probability: Step-by-Step Guide

  • Thread starter Thread starter Niamh1
  • Start date Start date
  • Tags Tags
    Probability Proof
AI Thread Summary
The discussion revolves around proving a probability statement using the binomial theorem, specifically the equation (x+y)^n = Σ(n choose k)x^ky^(n-k). Participants explore the implications of setting x=1 and y=1, leading to the result 2^n. A key point made is the derivation of the sum S = Σ(r * (n choose r)), which connects to the binomial theorem and ultimately shows that 2S = n2^n. The conversation emphasizes the importance of understanding the relationship between the summation and the binomial theorem to achieve the proof. The discussion concludes with a focus on the method of connecting these mathematical concepts.
Niamh1
Messages
4
Reaction score
0
View attachment 6593

Prove the following

I literally have no idea where to start or what to do.
 

Attachments

  • Screenshot (189).png
    Screenshot (189).png
    2.4 KB · Views: 101
Mathematics news on Phys.org
Hi there. :)

The binomial theorem states that $$(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^ky^{n-k}$$. What happens if you let $x=1$ and $y=1$?
 
Jameson said:
Hi there. :)

The binomial theorem states that $$(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^ky^{n-k}$$. What happens if you let $x=1$ and $y=1$?

Doesn't really help me to be honest.
 
Hi Niamh,

Let $S = \sum_{r = 1}^n r\binom{n}{r}$. Then

$$S = 0\binom{n}{0} + 1\binom{n}{1} + 2\binom{n}{2} + \cdots + (n-1)\binom{n}{n-1} + n\binom{n}{n}$$
$$S = n\binom{n}{n} + (n-1)\binom{n}{n-1} + \cdots + 1\binom{n}{1} + 0\binom{n}{0}$$

Using the fact that $\binom{n}{r} = \binom{n}{n-r}$ for $0 \le r \le n$, add the two equations column-wise, then use the binomial theorem, to obtain $2S = n2^n$. Dividing by $2$ yields the result.
 
Here's another easy proof using calculus:

$$(x+1)^n=\sum_{r=0}^n{{n}\choose{r}}x^r$$

Take derivatives:

$$n(x+1)^{n-1}=\sum_{r=1}^nr{{n}\choose{r}}x^{r-1}$$

Eavaluate at x=1:

$$n2^{n-1}=\sum_{r=1}^nr{{n}\choose{r}}=\sum_{r=0}^nr{{n}\choose{r}}$$
 
Niamh said:
Doesn't really help me to be honest.

Hi Niamh,

My post and the two below show the main connection you need, that is how to connect the summation sign to something resembling $2^n$. At first glance this is a quite strange equation and the key is the binomial theorem as I mentioned. If you let $x=1$ and $y=1$ you get:

$$(x+y)^n =(1+1)^n=2^n= \sum_{k=0}^{n}\binom{n}{k}x^ky^{n-k}=\sum_{k=0}^{n}\binom{n}{k}1^k1^{n-k}=\sum_{k=0}^{n}\binom{n}{k}$$

Using this trick we have now connected $2^n$ to something with this summation. That was the idea behind my post. :)

What have you tried?
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top