Probability

8.

If two people pick a number from 1 to 80, then there are 79 possible adjacent pairs that these numbers could go in. For each of these pairs, there are two options (one with person a picking the lower number and one with person b picking the lower number). Thus there are 158 possible pairs. The chance that person 1 will pick any given number is 1/80. The chance that person two will pick any given number is 1/80. Thus the probability that they have picked adjacent number is (1/80)2x158 = 0.0246875

1 in 13 chance

Naturally if left undisturbed not all that high. But this is one of those things that changes wildly based on other variables. Are there kids in the house, do you live in a "bad" part of town, do you live in a country experiencing war or riots, is the wiring in the house up to code, is the house in an open field, is the house in an area known for wildfires, how much rain has there been in recent times, are any members of the house convicted arsonists, ok by now if you're still reading this you should get the idea.

That being said, if this is secretly an insurance question. Yes you should get homeowners coverage that covers fire.

1\12

1/6

The probability of drawing any single card in a deck of 52 cards is 1 in 52. It does not matter if you replace it or not, unless you intend to make multiple draws.

In the multiple draw case, each subsequent probability increases; 1 in 51, then 1 in 50, then 1 in 49, etc.

As an example, the probability of drawing the seven of diamonds in five draws from a standard deck is 1 in 52 plus 1 in 51 plus 1 in 50 plus 1 in 49 plus 1 in 48. This adds up to about 1 in 10.

360.

There are 6 letters, so there are 6! (=720) different permutations of 6 letters.

However, since the two 'o's are indistinguishable, it is necessary to divide the total number of permutations by the number of permutations of the letter 'o's - 2! = 2

Thus 6! ÷ 2! = 360

The electron cloud. The atomic radius roughly describes the distance from the nucleus to the electron cloud.

Sampling error leads to random error. Sampling bias leads to systematic error.

Probabilities go from 0 to 1. 1 means the event will definitely happen. 0 means it's impossible for it to happen. 0.5 means it's just as likely to happen as not. Once you are aware of this scale, it is clear that an event with probability 0.95 is more likely to happen than an event with probability 0.10.

(assuming that the probability of having a girl or a boy is 50/50)

Looking from beforehand, the probability of having three boys then a girl is the probability of each of these events happening multiplied together. That is 50% x 50% x 50% x 50% or 0.54 This would mean that the chance of having a girl after three boys is 0.0625.

If you've already had the three boys though, it is a different story. The point is that previous experiences do not affect future ones; probability has no memory. Thus the probability of having a girl next is 50%, regardless of if you've had boys or girls in the past. To think otherwise is known as the gambler's fallacy, where a gambler says "black has come up 4 times in a row, it must be red next" even though the chance of red is always 50%

They are b, l, p, r, t and y (although the y acts as a vowel).

A fully extended standard windsock, per FAA standards, means the wind is at least 15 knots or 17 MPH.

Possible results when rolling the first cube = 6

Possible results when rolling the second cube = 6

Possible results when rolling two cubes = 6 x 6 = 36

How many of these 36 possibilities are snake-eyes ? Only 1 .

Probability of snake-eyes on the first roll = 1/36

Probability of snake-eyes on the second roll = 1/36

Probability of snake-eyes on both rolls = 1/36 x 1/36 = 1/1,296 = 0.077 percent

Odds = (1 in 1,296) or (1,295 to 1) against it.

The probability of failure.

Actually, the phrases used have very confusing connotations. When studying the risk of people being killed by lightning strikes, a "success" or the "desired outcome" is death caused by lightning. I somehow doubt that the person struck would consider it a desired outcome or a success!

Probability is the likelihood, expressed in numerical or ratiometric terms, that an event will occur. A probability of 1 means that the event will occur. A probability of 0 means that the event will not occur. A probability of 0.5 means that the likelihood of the event occurring is equal to the likelihood of it not occurring. For instance, a fair coin has a 0.5 probability of being heads, and a 0.5 probability of being tails.

Defined formally, probability is the number of permutations of the desired outcome divided by the number of permutations of all possible outcomes. Take a standard six-sided die, for instance. There are six permutations. One of them is a 1, so the probability of rolling a 1 is 1 in 6, or about 0.1667.

Probability is not assured. If you roll a die 600 times, you will not necessarily get 100 1's. Over the long run, you will approach that outcome, but each trial will have different results. This is the difference between theoretical probability and experimental probability - theoretical being the mathematical estimate - experimental being the observed results.

The probability of drawing a diamond from a standard 52-card poker deck without jokers is 13/52, or 1/4.

The probability of drawing a second diamond at that point would then be 12/51, for an overall probability of 12/212, or 3/53. This amounts to about a 5.88% chance.

It is approx 0.44

1/2 or 5/10

Standard deviation is the square root of the sum of the squares of the deviations of each item from the mean, i.e. the square root of the variance. In order to increase the standard deviation, therefore, you need to increase the average deviation from the mean.

There are many ways to do this. One is to move each item further away from the mean. For example, take the set [2, 4, 4, 4, 5, 5, 7, 9]. It has a mean of 5 and a standard deviation of 2.14. Multiply each item by 2.2 and subtract 5, giving the set [-1.3, 2.9, 2.9, 2.9, 5, 5, 9.2, 13.4], effectively moving each item 10% further away from the mean. This still has a mean of 5, but the standard deviation is 4.49.

A higher standard deviation means that the data are fluctuating more widely with respect to the mean. It could mean there are some bad samples, or it could simply mean that the data are not as tightly bound to the mean as anticipated. An unexpected standard deviation should be evaluated, using more robust analyses techniques, so as to differentiate between the various explanations. This is an expected part of error analysis, without which an analysis is incomplete.

9

Because the whole population might be too large to sample.

A good example is the population of the world. At nearly 7 billion people, it would be unrealistic to sample each person to determine some factor that you are looking at. Generally, we sample a subset of the population, taking into account differences (or errors) that might result, in this case, regional and cultural, in order to estimate the behavior of the larger population.