2
votes
1answer
23 views

Probability of two independent random variables being equal

Assume that $X$ and $Y$ are two independent random variables that follow the binomial distribution of parameters $p$ (the probability of one success) and $n$ (the number of trials). I was wondering ...
1
vote
1answer
43 views

Binomial Distribution - independence

I have the following problem that I'm stuck on a few parts. ...
0
votes
0answers
74 views

Solving sample size of hypergeometric distribution given a specific probability

I am trying to figure out how to calculate the sample size of a hypergeometric distribution, given a population, population successes, and probability. Here is the initial formula: ...
1
vote
1answer
38 views

In a lottery of $90$ numbers a man adds extra $1,2,3$

Consider a lottery where $5$ balls are chosen randomly among $90$ balls numbered from $1$ to $90$. A man cheats adding to the $90$ balls, before the draw, three more balls numbered $1,2,3$. We say ...
1
vote
0answers
26 views

Gambler's ruin, probability of loss, infinite turns

A gambler starts with $\$1$ and bets $\$1$ every turn of a game, where he has the probability $p$ to obtain $\$2$ and $1-p$ to obtain nothing. If $p<1/2$, what is the probability he will eventually ...
0
votes
1answer
23 views

Binomials for getting probability of standard deviation

I have the following problem which I am stuck on the second part. Suppose that $30\%$ of all students who have to buy a text for a particular course want a new copy whereas the other $70\%$ want a ...
0
votes
1answer
29 views

Binomial Distribution for defects

I'm stuck on the following problem: A batch of components has arrived at a distributor. The batch can be characterized only if the proportion of defective components is at most 0.10. ...
2
votes
3answers
58 views

Probability of 5 cards drawn from shuffled deck

Five cards are drawn from a shuffled deck with $52$ cards. Find the probability that a) four cards are aces b) four cards are aces and the other is a king c) three cards are tens and ...
2
votes
1answer
55 views

Question about balls in urns

Suppose there are $n$ balls in an urn, and $r$ of them are red. I select $m$ balls from this urn at random. What is the probability that at least $k$ of them are red? $m$ must be less than $n$, but ...
1
vote
2answers
76 views

DICE - Rolling at least *k* on *n* six-sided dice - with a twist!

I am putting together a table of dice probabilities for a project I am working on and have found myself intimidated by a little "special case" I'm trying to work with. For determining the probability ...
1
vote
1answer
41 views

Lower bound functional binomial r.v.

I am trying to find a bound of the type $\mathbb{E}(|B-\frac{N}{2}|) \geq C \sqrt{N}$ Where $B$ is a binomial variable with parameters $(N,\frac{1}{2})$. The bound doesn't need to be very tight in ...
0
votes
1answer
44 views

Convexity of Binomial Term

I am reading a book on the probabilistic method, and the following claim was made: $\dbinom{y}{n}$ is convex. Why is this the case?
1
vote
2answers
42 views

Probability problem with binomial/multinomial distribution

Mary knows the answers to $20$ of the $25$ multiple choice questions on the Psychology $101$ exam, but she has skipped several of the lectures, she must take random guesses for the other five. ...
0
votes
2answers
53 views

Pascal's Triangle Proof

Trying to determine a formula for the sum of the entries of the $n$th row of Pascal’s triangle, for any natural number $n$. Any proof will do as I have to determine $3$ different proofs. - So far, ...
0
votes
2answers
56 views

How Do You Calculate Probabilities of Random Events Occuring in Sequence?

So I have a series: $f(x_{n+1})=x_n \pm t$ and $f(x_0)=W$ What I'd like to calculate is the probability in terms of $t$ and $W$ (assuming they're any constant $W>t$) that any $f(x_q)=0$ for all ...
8
votes
2answers
164 views

An extrasensory perception strategy :-)

Inspired by classical Joseph Banks Rhine experiments demonstrating an extrasensory perception (see, for instance, the beginning of the respective chapter of Jeffrey Mishlove book “The Roots of ...
2
votes
1answer
116 views

Bins and balls model - filling first bins [close]

We have $n$ bins and $m$ balls. I want to compute the probability that in the first $k$ bins, $q$ of them will be non-empty. I can throw $m$ balls into $n$ bins in $n^m$ ways. Using Stirling ...
0
votes
3answers
69 views

Probability of choosing coins from a bag (why doesn't binomial coefficient work?)

Studying for my exam and would appreciate some help. I have a bag with 2 pennies, 1 nickel and 1 dime. I pick 2 at random. The solutions say: Pr(PP) = $\frac{2}{4} \times \frac{1}{3}$ = ...
1
vote
1answer
54 views

Simple Random Walk

How to find: $\lim\limits_{N \to \infty}\sum\limits_{m=0}^Nu_m$ where $u_m$=${2m \choose m}p^mq^m$ I know there are two cases to consider depending if $p$ and $q$ are equal or not. I should probably ...
0
votes
2answers
65 views

generating function and binomial distribution - counting

I am trying to understand generating function. I have the following problem: There are 50 students in the International Mathematical Olympiad (IMO) training programme. 6 of them are to be selected to ...
0
votes
2answers
27 views

negative binomial distribution problem

Find the probability that you find 2 defective tires before 4 good ones. There is a chance of a tire being defective at a rate of 5%. From my understanding with the negative binomial distribution we ...
0
votes
0answers
29 views

Analytical solution for binomial equation

Suppose that the random variable $X \sim \operatorname{Binomial}_{n,p}$, and suppose we have $p' \in [0,1]$. I have been asked to solve for the least $n$ such that $P(X \leq 2) = p'$. It was ...
9
votes
3answers
411 views

Prove the lecturer is a liar…

I was given this puzzle: At the end of the seminar, the lecturer waited outside to greet the attendees. The first three seen leaving were all women. The lecturer noted " assuming the attendees are ...
1
vote
1answer
44 views

Binomial distribution . Heads and Tails

Consider a coin with P(Heads) = 2/ 3 . We toss this coin 100 times (assume that the tosses are independent). Determine the probability that we get exactly 45 tails out of the 100 tosses. First, ...
2
votes
0answers
42 views

Upper bound for tail of binomial expansion

Let $P,R,T$ be integer constants with $PR$ much greater than $T$. Suppose I flip a coin $PR$ times, each time (independent of other times) getting heads with probability $1/P$. The probability that I ...
1
vote
1answer
350 views

binomial distribution(overbooking plane tickets)

I am having trouble with binomial distribution and this problem: an airplane has 200 seats, but 202 tickets are sold. Assume passengers do not show up with a probability of .03 independently. What is ...
2
votes
2answers
46 views

Binomial distribution false reasoning

While reading the answer of a previous question Binomial Distribution Question (Exactly/At Least $x$ Trials for Success), it got me thinking a little. I know the reasoning must be flawed somewhere, ...
1
vote
1answer
61 views

Change of variable in an infinite sum

I'm currently trying to understand a derivation from WolframMathWorlds. I got to step 6 where a change of variable happens. You can see the equation here. I understand everything except how they get ...
0
votes
1answer
88 views

$P(X-np>n\varepsilon)\leq E\{e^{\lambda \cdot (X-np-n\varepsilon) }\}$

For $X \mathtt{\sim} \text{Bin}(n,p), \lambda > 0, \varepsilon > 0$, how do you show the following? $$P(X-np>n\varepsilon)\leq E\{e^{\lambda \cdot (X-np-n\varepsilon) }\}$$ Unless I made some ...
4
votes
1answer
85 views

Simplify $\sum_{k=0}^n \frac{1}{k!(n-k!)}.$

Is there a way to simplify the expression $$\sum_{k=0}^n \frac{1}{k!(n-k)!}?$$ This came up when I was trying to determine $\mathbb{P}(X+Y =r)$ given a joint mass probability $$m_{X,Y}(j,k) = ...
2
votes
0answers
110 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
5
votes
3answers
409 views

How many ways to choose $k$ out of $n$ numbers with exactly/at least $m$ consecutive numbers?

How many ways to choose $k$ out of $n$ numbers is a standard problem in undergraduate probability theory that has the binomial coefficient as its solution. An example would be lottery games were you ...
1
vote
1answer
65 views

Binomial Approximation. Calculate $n$

I have the next question: If it is known that $P(X \geq300)=0.3$ and $X \sim \mathop{Bin}(n,0.2)$ How can one estimate $n$?
1
vote
0answers
81 views

Feller vol. I, probability of 'r' successes in binomial process

This question is about the calculation of probability of at least 'r' successes in a binomial process given in the page 151 of feller:intro to probability:vol-1. Before deriving equation 3.5, text ...
0
votes
2answers
425 views

Probability: Biased Die

Suppose we have three 6-sided die that all share the same common bias: For a single dice: let the probability of rolling a 2 or $P(2) = 2{\times}P(1$), let the probability of rolling a 3 or $P(3) = ...
3
votes
2answers
420 views

German tank problem, simple derivation [duplicate]

I was reading the recent question on the German tank problem, and had trouble with one of the derivations in this section. $$\sum_{m=k}^N m \frac{\binom{m-1}{k-1}}{\binom N k} = ...
1
vote
1answer
68 views

Binomial Function and the hypergeometric function

So lets define: $$h(r|M,N,n) = \frac{\binom{M}{r} \binom{N-M}{n-r}}{N \choose n}$$ Having the usual $\Gamma$ function we define the hypergeometric function: $$F(a,b,c;t) = ...
1
vote
1answer
47 views

Problem with binomial sumation

I am trying to solve the following summation: $\sum_{k=0}^{M-1} {M-1 \choose k} \alpha^{k} (1-\alpha)^{M-1-k} u(k)$ where: $u(k) = 1$ , if $0 \le k < j$ $u(k) = (1-\frac{j}{k+1})$ , if $M-1 \ge ...
3
votes
1answer
223 views

Average absolute value of sum with Rademacher random variables

Let $a_1, \ldots, a_n $ be independent Rademacher random variables with distribution $P(a_i=1) = P(a_i=-1) = \frac 12$. Estimate from below $$E \left|\sum_{i=1}^n a_i\right|.$$ I've reduced this ...
3
votes
1answer
501 views

Binomial theorem in probability

We know according to binomial probability theorem , $$P= \binom{n}{r} p^r (1-p)^{n-r} \tag{1}$$ Now If I flip a coin 10 times and want to get the probability for 4 heads then we get according to the ...
2
votes
0answers
167 views

Approximating hypergeometric distribution with poisson

I'm am currently trying to show that the hypergeometric distribution converges to the Poisson distribution. $$ \lim_{n,r,s \to \infty, \frac{n \cdot r}{r+s} \to \lambda} \frac{\binom{r}{k} ...
0
votes
1answer
196 views

Sum of binomial probabilities

One of my friends is building a game where the player will get questions from 6 different categories. Each category has a total of 50 questions. A single game consists of answering one question from ...
1
vote
3answers
51 views

Proof that $\sum_{k=0}^m \binom{m}{k}\frac{1}{k+1} = \frac{2^{m+1}-1}{m+1}$ [duplicate]

Recently I needed to compute $E[\frac{1}{X+1}]$ where $X\sim Bin(m, \frac 1 2)$. While expanding, I came across the sum $\sum_{k=0}^m \binom{m}{k}\frac{1}{k+1}$, which I was unable to solve. Plugging ...
4
votes
2answers
149 views

Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$

I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$ I know that $$2^n = ...
6
votes
2answers
189 views

Binomial probability with summation

Show that $$\sum_{k=0}^{m} \frac{m!(n-k)!}{n!(m-k)!} = \frac{n+1}{n-m+1}$$ Attempt: It becomes: $$\sum_{k=0}^{m } \frac{\binom{m}{k}}{\binom{n}{k}}$$ Telescoping, pairing, binomial theorem don't ...
7
votes
3answers
85 views

Show that ${-n \choose i} = (-1)^i{n+i-1 \choose i} $

Show that ${-n \choose i} = (-1)^i{n+i-1 \choose i} $. This is a homework exercise I have to make and I just cant get started on it. The problem lies with the $-n$. Using the definition I get: $${-n ...
0
votes
2answers
47 views

standard deviation of a certain distribution

If I have a list of N outcomes of drawing a number from the set {-1\$,+1\$}, and I know that the probability of getting (in a single draw) (-1\$) is p, and probability of getting (in a single draw) ...
3
votes
3answers
98 views

Probability someone's phone will ring during a movie?

Trying to figure out what the probability is that in a room of 200 people what the probability that at least one will get a phone call during a certain time window... In this case 2 hours ...
-1
votes
1answer
44 views

Formula issues when working out chances of getting certain marks [duplicate]

$$P(X = k) = \binom{N}{k} (0.5)^k (0.5)^{N-k} = \binom{N}{k} (0.5)^N$$ Using formula above, I have got the following results for chances for getting certain percentage on a $50$ question paper, each ...
0
votes
3answers
134 views

How to correctly write a binomial distribution for a $50$ questions exam [duplicate]

Using binomial distribution I want to know what is the chance of getting $70\%$ or greater in a $50$ question exam, each question having a true/false option to select from. What is the correct formula ...