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Probability
The probability of a certain event is a number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences. In mathematics, it is a measure of how often an event will happen and is the basis of statistics.
14,643 Questions
How many marbles can you fit in a jar?
That depends on the size of the marbles, as well as the size and shape of the jar.
The Central Limit Theorem is important in statistics because?
According to the central limit theorem, as the sample size gets larger, the sampling distribution becomes closer to the Gaussian (Normal) regardless of the distribution of the original population. Equivalently, the sampling distribution of the means of a number of samples also becomes closer to the Gaussian distribution. This is the justification for using the Gaussian distribution for statistical procedures such as estimation and hypothesis testing.
What is the element of the sample space for a coin then rolling a die?
It is an ordered pair of the form (A, n) where A is the outcome of the tossed coin (H or T) and n is the outcome of the rolled die (1, 2, 3, 4, 5, 6).
Three fair dice are tossed What is the probability that at least two of them show that same face?
Total number of outcomes = 6 x 6 x 6 = 216
Number of ways that are successful:
1 1 -
1 - 1
- 1 1
1 1 1
2 2 -
2 - 2
- 2 2
2 2 2
3 3 -
3 - 3
- 3 3
3 3 3
etc.
Four ways for each number 1 thru 6 = 24 possible successes. But for each of the one's with a dash (-), there are 5 outcomes (121, 131, 141, 151, 161, 211, 311,411,...etc), so the real number of ways for this to happen is 16 ways for each number: 6 x 16 = 96.
Probability of success = 96/216 = 4/9 = 44.4% (rounded)
What are the chances of dealing out cards that have been shuffled well in exact order?
The odds are... remote. This is a factorial problem to calculate the answer you take how many options the first "dealt" card has of being. In this case the first card has a 1-in-52 chance of being the Ace of Spades (the "highest" card in a deck.) The next card has a 1-in-51 chance of being the Two of Spades, and so on. You calculate the odds of this happening my multiplying 52 by 51.
So, the odds are 52 times 51 times 50 times 49 times 48 times... yadda yadda yadda until you get to one. The answer?
The odd of taking a shuffled deck of cards, reshuffling it, and the dealing the cards back in the correct order are 1 in 806,581,751,709,439x10 to the 54th power. (septendecillion.)
Or 80 unvigintillion. (80 followed by 66 zeroes). For the sake of comparison, conservative estimates put the number of stars in our galaxy at 100 Billion (100 followed by 9 zeroes) and the number of galaxies in the universe at the same number, making the number of potential stars in the universe at 10 sextilion (a 10 followed by 21 zeroes.)
A sample.
What is the critical z value thst corresponds to a confidence level of 88 percent?
The answer will depend on whether the critical region is one-tailed or two-tailed.
What is the probability of getting a diamond and five without replacement?
The answer depends on how many cards are drawn.
Total marbles in the bag = 10
Number of red ones = 3
Probability of pulling a red one on the first draw = 3/10 = 0.3
Total marbles remaining after the first draw = 9
Number of green ones = 5
Probability of pulling a green one after a red one has been withdrawn = 5/9
Probability of both outcomes = (3/10) x (5/9) = (15/90) = 1/6 = (16 and 2/3) percent.
Primes are 2, 3, 5 and 7 which form 6 "prime pairs". There are 45 possible pairs in the box so the odds are 39 to 6 against ie 13/2, a probability of two-fifteenths or 13.3% or 0.13 recurring.
When is the median the most appropriate center for a distribution?
The median is the most appropriate center when the distribution is very skewed or if there are many outliers.
How many standard deviations is the first quartile away from the mean on a Normal distribution?
0.674 sd.
What is the probability of rolling a sum of 7 or a sum of 11?
If two fair number cubes are rolled, the answers are:
Prob(7) = 6/36 = 1/6
Prob(11) = 2/36 = 1/18.
When rolling two dice which outcome is more likely a sum of 3 or sum of 5?
For 3 you need 1+2, but for 5 you can have 1+4 or 2+3 so you have twice as many chances for 5.
Empirical
Can the standard deviation or variance be negative?
No, a standard deviation or variance does not have a negative sign. The reason for this is that the deviations from the mean are squared in the formula. Deviations are squared to get rid of signs.
In Absolute mean deviation, sum of the deviations is taken ignoring the signs, but there is no justification for doing so. (deviations are not squared here)
There is no simple answer to the question because the children's genders are not independent events. They depend on the parents' ages and their genes.
However, if you assume that they are independent events then, given that the probability of a boy is approx 0.52 in all cases, the overall probability is 0.0624.
Part1: Finding probability of getting sum as a perfect square.
Maximum sum of both the dice is (6+6) equal to 12.
Up to 12, the perfect squares are: 1, 4 and 9.
Getting a sum of 1 from two dice is not possible. So, we are left with 4 and 9.
To get 4, the combination can be: (2,2) or (1,3) or (3,1). This means, to get the sum as 4, the probability is [3/36].
To get 9, the combination can be: (3,6) or (6,3) or (5,4) or (4,5). This means, to get the sum as 9, the probability is [4/36].
Therefore,the total probability of getting the sum as a perfect square is: [(3/36)+(4/36)]=[7/36].
Part2: Finding the probability of getting sum as an even number.
The possible even numbers can be 2, 4, 6, 8, 10 and 12. But, as 4 is already considered in part1, it should be ignored in this case.
The probability of getting sum as 2 is: [1/36]
The probability of getting sum as 6 is: [5/36]
The probability of getting sum as 8 is: [5/36]
The probability of getting sum as 10 is: [3/36]
By adding all the above, the probability of getting sum as an even number (ignoring 4) is:
[(1/36)+(5/36)+(5/36)+(3/36)]=[14/36].
From part 1 and part 2, we get the total probability as [(7/36)+(14/36)]=[7/12]=0.583333.
What is the meaning of expected value in a probability distribution?
The expected value is the average of a probability distribution. It is the value that can be expected to occur on the average, in the long run.
