Probability

They are both concepts of a branch of mathematics that is called statistics.

For a single roll of a pair of fair dice, the answer is 1/36.

Tables of the cumulative probability distribution of the standard normal distribution (mean = 0, variance = 1) are readily available. Almost all textbooks on statistics will contain one and there are several sources on the net. For each value of z, the table gives Φ(z) = prob(Z < z). The tables usually gives value of z in steps of 0.01 for z ≥ 0. For a particular value of z, the height of the probability density function is approximately 100*[Φ(z+0.01) - Φ(z)].

As mentioned above, the tables give figures for z ≥ 0. For z < 0 you simply use the symmetry of the normal distribution.

It is 17C5*(1/16)5*(15/16)12 = 0.00272 approx.

There are two main methods: theoretical and empirical.

The first of these is used when there is a relatively simple model for the possible outcomes of a trial. For example, if you roll a fair die, laws of physics suggest that each of the six faces is equally likely to end up on the top face. The probability for each of the six numbers is, therefore, 1/6.

The second method is used when there is no satisfactory model based purely on theory: if you have a loaded die, for example. It may, just about, be possibly to analyse the physical properties of the mass distribution within the coin and develop an appropriate model for the outcome of a throw. However, it is simpler to use the second method. Given the chance, roll the die again and again (and again and again and ... ) and record the outcomes. The probability of any particular outcome is the proportion of the trials that result in that particular event. Thus, if a loaded die comes up 6 fifty times out of 200 throws, then the probability of throwing a 6 is 50/200 = 0.25.

Rolling tables can be purchased on various websites on the internet. Some of the stores are Grainger, Amazon and Garner Tackle. It may be worth comparing products between the sites to ensure you get the best deal.

There are many companies that sell Spinner bikes. You can purchase them from Spinning themselves, or buy them from sites like Amazon or various athletic stores.

P-rod might have quit PLan B

Pictures show that at Portland Street League 2013 he was using a blank board. Yes all companies provide blank boards but his wasn't a PLan B blank board.

On the list of sponsors he had; Mountain Dew, Grizzly, Target, Nike, etc. were there but Plan B wasn't on the list anymore. Friends told me that he is starting a new company. I don't know if he really quit Plan B. But we'll find out soon enough.

Playing cards have been around in China since the 600s. The 52 card deck originated in about 1400 by Arabs. Four suits already existed, and I'm not sure anybody knows exactly why they chose to have 13 in a suit, but since it goes 1 through 10 and then a prince (jack), a queen and a king it makes some kind of sense. So 4 times 13 equals 52.

You have a good chance on getting two questions right of six true -or - false questions. The chances are high because you have a few questions.

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Let N be the number of trials for the simulation, chosen in advance.

Let U be a source of pseudorandom uniform numbers on the unit interval [0,1].

We simulate guessing on one question by taking a number from U and comparing it with 0.5. If it's greater then we call it a 'success' on the question.

trial_count = 0

success_count = 0

while True :

increment trial_count

question_success_count = 0

do six times :

if U > 0.5 then increment question_success_count

if question_success_count = 2

then increment success_count

if trial_count = N then quit loop

probability = success_count / N

This stupid system discards indentation. I regret you will probably need to reconstruct that.

Depending on the number of samples involved, you either simulate all possible samples or you simulate taking a large number of samples. The distribution of he sampling statistic can be calculated from these.

Here's a Python script that demonstrates how to do this in a simple form. Suppose you want to experiment with the sampling distribution of the sample mean for samples of size 5 drawn from a normal population with mean 0 and standard deviation 1. There are, of course, mathematical results that establish the exact sample distribution of this statistic. Let's pretend we don't know that. Usually it's necessary to generate many, many samples to establish a sampling distribution. For the purposes of this exercise this code generates only 20.

from scipy.stats import norm

N = norm ( 0., 1. ) ## create source of N(0,1) deviates

x_bars = [ ] ## create place to hold sample x_bar values

for s in range ( 20 ) : ## run experiment 20 times

x_bars . append ( sum ( N . rvs ( 5 ) ) / 5. ) ## make & store sample x_bar values

x_bars . sort ( ) ## form the so-called order statistics for the sample

for i, x_bar in enumerate ( x_bars ) : ## print empirical distribution function

print '%.2f %.2f' % ( ( i + 1 ) / 20., x_bar, )

When I ran this code I got the following:

0.05 -1.65

0.10 -0.28

0.15 -0.25

0.20 -0.10

0.25 -0.09

0.30 -0.08

0.35 0.01

0.40 0.05

0.45 0.10

0.50 0.12

0.55 0.13

0.60 0.19

0.65 0.41

0.70 0.46

0.75 0.50

0.80 0.55

0.85 0.67

0.90 0.79

0.95 0.94

1.00 1.05

If you plot these using the column on the left as the x-axis you will get the so-called empirical distribution function. The sample values can also be used in a variety of ways to obtain estimates of the sampling probability density.

Please see the link for the basics.

If the die is rolled often enough, the event is a certainty - probability = 1.

For a single roll, the probability is 1/2.

Probably not........:-)

Uniform probability can refer to a discrete probability distribution for which each outcome has the same probability.

For a continuous distribution, it requires that the probability of the outcome is directly proportional to the range of values in the desired outcome (compared to the total range).

The opposite of getting at most two heads is getting three heads. The probability of getting three heads is (1/2)^2, which is 1/8. The probability of getting at most two heads is then 1 - 1/8 which is 7/8.

The answer depends on what the experiment is!

6

The answer depends on the experiment: how many coins are tossed, how often, how many dice are rolled, how often.

It is 0.999531889

Prob(at least one Blue if you spin it 5 times)

= 1 - Prob(No Blue in 5 spins)

= 1 - (3/4)5

= 1 - 243/1024

= 781/1024 = 0.7627 approx.

If you can enumerate the outcome space into equally likely events, then it is the number of outcomes that are favourable (in which the event occurs) divided by the total number of outcomes.

If dealt from a randomly shuffled pack it is 0.0399, approx.

If dealt from a randomly shuffled pack it is 0.0399, approx.

If dealt from a randomly shuffled pack it is 0.0399, approx.

If dealt from a randomly shuffled pack it is 0.0399, approx.

It's unlikely to happen