Statistics

Number of ways that the thousand digit can be odd = 5 (1, 3, 5, 7, or 9)

Number of ways that the hundred digit can be even = 5 (0, 2, 4, 6, or 8)

These two digits are always different. Total number of possible pairs = 5 x 5 = 25.

For each pair, the tens digit can be any one of 8 digits (it can't be the same as

either the thousands or the hundreds) = 8 possibilities for the tens digit.

For each of those, the units digit can be any one of 7 digits (it can't be the same as

the thousands, the hundreds, or the tens) = 7 possibilities for the units digit.

Total number of different numbers that can be written according to these rules =

25 x 8 x 7 = 1,400

Even with no mistakes in a carefully conducted experiment, error is expected. That word error does not mean a mistake, it means that measurements can never find exactly the quantity being measured.

Suppose you measure the length of a table top, you might find it's 48.8 inches. Does that mean 48.800000 inches? no it does not, because you can't measure to an accuracy of one millionth of an inch without special equipment. Your experimental error is the difference between your measurement and the exact length of the table.

Here are three possible interpretations of the question, with answers:

A) How many combinations are possible when when rolling three identical regular dice simultaneously, if all the dice show an even number?

Answer: 10 (Originally given by Mehta matics)

... (2,2,2) -> 6

... (2,2,4) -> 8

... (2,2,6), (2,4,4) -> 10

... (2,4,6), (4,4,4) -> 12

... (2,6,6), 4,4,6) -> 14

... (4,6,6) -> 16

... (6,6,6)-> 18

B) How many combinations result in an even total when rolling three identical regular dice simultaneously?

Answer:28

... The totals must be between 3 and 18 inclusive.

... sum of 4: (1,1,2)

... sum of 6: (1,1,4), (1,2,3), (2,2,2)

... sum of 8: (1,1,6), (1,2,5), (1,3,4), (2,2,4), (2,3,3)

... sum of 10: (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4)

... sum of 12: (1,5,6), (2,4,6), (2,5,5), (3,3,6), (3,4,5), (4,4,4)

... sum of 14: (2,6,6), (3,5,6), (4,4,6), (4,5,5,)

... sum of 16: (4,6,6), (5,5,6)

... sum of 18: (6,6,6), for a total of 1+3+5+6+6+4+2+1 =28 combinations.

C) When rolling three regular dice, how many even totals are possible?

Answer: 8

... 4, 6, 8, 10, 12, 14, 16, 18.

Answer 1:
The odds are very easy to calculate. Simply divide the number of "valid" rolls against all possible rolls. For ease, you can write down all possible combination for the 2 dice.

1-1; 1-2; 1-3; 1-4....and so on, remember 1-4 and 4-1 are different rolls

There are 36 unique possible combination, and 6 of them are doubles, so that's 6/36 chances (and since 6 goes into 36, 6 times, this reduces to 1/6) or about 17%

Answer 2:

Another way to look at this problem, generically, is to assume we have an 'n' face dice. In most cases, dice have 6 faces (1, 2, 3, 4, 5, 6). But why not create a solution that works for any number of sides? Well, if we are trying to calculate the probability of rolling two dice (dice-1 and dice-2) of 'n' sides at the same time and having them turn up as doubles, only one of the dice really matters. Here's why. Dice-1 is guaranteed to land on a number 1-n. This will happen every time (on a fair dice, disregarding freak incidents). What we are trying to calculate is the probability that dice-2 will land on the SAME number as dice-1. Dice-2 can only land on one of 'n' values: 1, 2, 3, 4, 5, 6, ... , 'n'. For you non math folks, this just means it must land on a number from 1 to 'n' where 'n' is the number of sides on your dice. Out of all of the sides that dice-2 can PHYSICALLY land on, one of the sides MUST necessarily have the same as the value that dice-1 landed on. That is to say, if dice-1 landed on the value 3, there must be some chance that dice-2 will also land on the value 3. The probability of this occurring on a fair die is 1 divided by the total number of possible outcomes, which would be 'n'. So, really, there is a 1/n chance that dice-2 will land on the same number as dice-1. Thus, our probability for rolling doubles is simple 1/n. For our 6 sided dice example, our dice-1 lands on some value between 1 and 6 and there is a 1/6 chance dice-2 will match it.

The person in knowledge is a better person as he knows more about theprinciples that govern the material, metaphysical and spiritual worlds andhence fully adheres to those principles n gets better results.

The person in passion may have the same knowledge but in thehaste of acting misses out on some elements of that action and gets a lesserresult.

For example if I plant an Alphanso mango seed and nurture it organically - ittakes 7 plus years before I get to enjoy the ripened and most tasty fruit. Andafter that every alternate year!

If I have the passionate nature, I will feverishly add chemicals etc n get toenjoy the ripened fruit in 4 odd years and every year there after- but thenatural taste is lost- its not so tasty!

The person in goodness/knowledge is calm n collected in his actions anddoes them in proper sequence and in knowledge and thereby obtains full resultswhere as the person in passion is in a hurry and will definitely miss out on afew sequential actions thereby losing out on achieving the correct result.

So to get best results one must act in full knowledge- calmly.

There is one rider though about the later type of person- that person could bein ignorance!

The external manisfestation of an ignorant person and a person in fullknowledge appears to be the same- the results are totally different.

For example the person sweeping the floor in goodness or in ignorance appear Tobe doing the same thing- sweeping the floor slowly but the person in knowledgeis sweeping away the dirt in one direction till it is collected into a dust panwhere as the person in ignorance is sweeping slowly but pushing some dirt backinto the swept area and doing the sweeping action again and again and missingout on some dirt- the result one sees in the dust collected and what remains on the floor!

I prefer to be the person who takes time to live a slower paced life but inknowledge- I will definitely achieve fulfillment and greater self satisfaction.

That is an intelligent way of living.

kanai

Disclaimer: This is not an easy answer because a lot depends on the sample size, distribution of population, etc. But for the sake of this I will assume that we are talking about a normally-distributed population and a sufficiently large sample size.

You want to test the hypothesis that smokers need less sleep than the general public. You are told that the general public needs an average of 7.7 hours of sleep. So:

Avgpop = 7.7

The hypothesis that you are testing (H1) is that smokers need less sleep:

Avgsmoker < Avgpopulation or Avgsmoker < 7.7 hours

Your null hypothesis (Ho) would be that there is no difference between the two groups OR that smokers need MORE sleep than the general population:

Avgpopulation >= Avgsmoker or Avgsmoker <= 7.7 hours

You test that hypothesis by taking a sample of X smokers and comparing their average number of sleep to that of your population. You then test to see whether any differences you observe would likely have been found due to chance or because the sample is different from the population. The more smokers that you randomly sample, the more likely it will be that a given difference will be due to smokers need a different amount of sleep than the general public.

Lets say that you sampled 200 smokers and their average night of sleep was 7.55 hours with a standard deviation of 1.0. Now we can calculate the test statistic. Because we are assuming a normal population and the sample size is greater than 30, we can use a one-sample z-test (it is one sample because we are taking one sample and comparing it to a known mean).

Z = (Sample Mean - Population Mean)/(Standard Deviation of Sample/Square Root of Sample Size)

Z = (7.55 - 7.7)/(1.0/Sqrt(200))

Z = -.15/.0707

Z = -2.12

We now compare our test statistic to a critical value table. (one for the z-statistic can be found here: http://math2.org/math/stat/distributions/z-dist.htm

That Z-statistic corresponds to a p-value of .017. This means that the probability of randomly drawing a sample with this observed average assuming the null hypothesis is true is .017 or 1.7%. At a confidence level of 98.3% and below you would reject the null hypothesis. THIS DOES NOT MEAN WITH CERTAINTY THAT SMOKERS NEED LESS SLEEP THAN THE GENERAL PUBLIC, however the data would suggest that the hypothesis that smokers need less sleep is true (in this case).