6 votes
Accepted

In a game with 0.01 chance of survival, there are 100 participants, a specific player survives twice what are the odds?

The probability of a given player surviving twice (assuming survival is purely by chance and the two games are independent) is $.01^2 = .0001$, or 1:9999 odds. The probability of there being one ...
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  • 21.9k
4 votes

What is Galton's paradox?

The "paradox" here arises from sneaking in an implicit insinuation of independence that does not actually hold, which allows the argument to lead you to a wrong answer with a series of ...
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3 votes

Hazard function and survival analysis

Answer Given that your $\lambda(t)$ actually represents the instantaneous probability to die at time $t$ (the hazard), your function $F(t)$ computes the probability to die before a certain time $t$. ...
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3 votes

MNIST with a TWIST, no labels given, only probabilities

By the way you’ve set up the problem, perfect accuracy is impossible, and we must accept that, even if we know a digit to be a $1$, there’s only a $90\%$ chance of being red, and that’s the best we ...
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3 votes

Probability of selecting 3 students being in different classes

This can be calculated more simply as: $$P(Friend1 \in any\ class) \times P(Friend2 \notin class_{friend1}) \times P(Friend3 \notin class_{friend1} \& Friend3 \notin class_{friend2})$$ $P(Friend1 \...
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  • 521
2 votes

Betting on a sample from a known distribution

The mean is affected by outliers in the distribution, so does not inform us about the mid-point of the distribution. The median is not affected by outliers. Consider three example distributions and ...
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  • 521
1 vote

How to interpret coefficients for a binary DV in an OLS model and constant value meaningfully?

There are a few different ways to model binary outcomes with an OLS approach, but the basic idea is that the coefficients represent the change in probability that the dependent variable is 1, holding ...
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1 vote
Accepted

Joint distribution of longest run and number of runs in a sequence of $n$ coin tosses?

For the special case of $p=0.5$, the distribution is described by restricted integer compositions (see this post). $$ F(n,k,w)=\sum_{j=0}^k(-1)^j\binom{k}{j}\binom{n-jw-1}{k-1} $$ where $n$ is the ...
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  • 311
1 vote
Accepted

Betting on a sample from a known distribution

The median minimizes expected absolute distance between observations sampled from the distribution and a single value. That is: $$ \underset{m}{\text{min}}\text{ }\mathbb E\left[ \left\vert X - m\...
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1 vote

What is meaning of such notations in general?

$$\mathbb{E}_{(\mathbf{u}, \mathbf{\sigma}, \mathbf{Y}_0)\sim N(0,1)\otimes \mu_D\otimes N(0,1)}\left[ e^{\mathbf{\sigma} t}\mathbf{Y}_0\mathbf{u}\right]=\int\int\int e^{\mathbf{\sigma} t}\mathbf{Y}_0\...
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