Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. For questions about the theoretical footing of probability (especially using [tag:measure-theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

Probability is a numerical quantity which lies in the interval $[0, 1]$. According to the Bayesian view, it represents the belief about an event or preposition and it is interpreted as how likely it is for an event to occur, or of how likely it is for a proposition to be true. According to the frequentist view, it represents the relative frequency of a favourable event with respect to the sample space. Use this tag for basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantities. For questions about the theoretical footing of probability (especially using ), please ask under instead. For questions about specific probability distributions, please use .

95623 questions
123
votes
6 answers

Intuition behind Conditional Expectation

I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know. Let me get more specific. Let…
120
votes
8 answers

probability $2/4$ vs $3/6$

Recently I was asked the following in an interview: If you are a pretty good basketball player, and were betting on whether you could make $2$ out of $4$ or $3$ out of $6$ baskets, which would you take? I said anyone since ratio is same. Any…
zephyr
  • 1,029
  • 1
  • 8
  • 5
119
votes
4 answers

Could someone explain conditional independence?

My understanding right now is that an example of conditional independence would be: If two people live in the same city, the probability that person A gets home in time for dinner, and the probability that person B gets home in time for dinner are…
Ryan
  • 1,591
  • 4
  • 11
  • 18
118
votes
9 answers

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here. Randomly break a stick in five places. Question: What is the probability that the resulting…
116
votes
4 answers

What is the difference and relationship between the binomial and Bernoulli distributions?

How should I understand the difference or relationship between binomial and Bernoulli distribution?
user122358
  • 2,552
  • 5
  • 20
  • 32
113
votes
20 answers

In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?

In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives, by Leonard Mlodinow, Vintage Books, 2008. On p.107…
NotSuper
  • 1,753
  • 4
  • 16
  • 16
107
votes
9 answers

If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, would it still be considered unfair for $50$…
Adam Freymiller
  • 1,051
  • 2
  • 8
  • 7
105
votes
2 answers

Expectation of the maximum of gaussian random variables

Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large? If $F$ is the cumulative distribution function…
Chris Taylor
  • 27,485
  • 5
  • 79
  • 121
100
votes
11 answers

Mathematician vs. Computer: A Game

A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same range. If the numbers chosen share a prime factor,…
user139000
100
votes
8 answers

Is Bayes' Theorem really that interesting?

I have trouble understanding the massive importance that is afforded to Bayes' theorem in undergraduate courses in probability and popular science. From the purely mathematical point of view, I think it would be uncontroversial to say that Bayes'…
user78270
  • 3,718
  • 2
  • 18
  • 27
96
votes
19 answers

Should I put number combinations like 1111111 onto my lottery ticket?

Suppose the winning combination consists of $7$ digits, each digit randomly ranging from $0$ to $9$. So the probability of $1111111$, $3141592$ and $8174249$ are the same. But $1111111$ seems (to me) far less likely to be the lucky number than…
arax
  • 2,731
  • 1
  • 16
  • 24
96
votes
10 answers

Given a die, what is the probability that the second roll of a die will be less than the first roll?

If you are given a die and asked to roll it twice. What is the probability that the value of the second roll will be less than the value of the first roll?
Salman Paracha
  • 1,111
  • 1
  • 8
  • 11
95
votes
12 answers

Why do bell curves appear everywhere?

Why do most probability graphs show a bell curve? I've been wondering why... Is it just something natural, like the fibonacci sequence?
Shreyas Shridharan
  • 1,121
  • 1
  • 8
  • 15
94
votes
13 answers

Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
leava_sinus
  • 1,045
  • 3
  • 9
  • 4
94
votes
4 answers

The "pepperoni pizza problem"

This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely randomly. All pepperoni have the same radius $r$. A…
Ben S.
  • 475
  • 1
  • 5
  • 14