Probability

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Standard deviation is the square root of the variance; so if the variance is 64, the std dev is 8.

Hasty generalizing

If you toss a fair coin 250 times , about how many times should it land on tails?

For standard dice, there are 36 outcomes for rolling 2 dice. There are 3 ways to roll a 4; 3,1 1,3 & 2,2. The probability of sum of 4 is then 3/36 or 1/12 or 0.0833.

Only one. A normal, or Gaussian distribution is completely defined by its mean and variance. The standard normal has mean = 0 and variance = 1. There is no other parameter, so no other source of variability.

8/33

First calculate the probability of NOT getting a six. This probability is 5/6 x 5/6 ... x 5/6, i.e., (5/6) to the power 6. Then subtract 1 minus this probability. This gives you the probability of getting at least one six.

6 Of course - assuming a cube die.

There are other dice available (for playing D&D) which can have for example 4 (tetrahedral dice) or 20 (icosahedral dice) outcomes.

5 out of 12

If you roll a pair of six sided number cubes the probability of rolling two numbers whose sum is 6 is 5/36 or 0.1389. There are 5 ways to get 6: 1,5 & 5,1 & 4,2 & 2,4 & 3,3. There are 36 outcomes for a roll of 2 dice.

Probability of Jack being drawn is 4/52 since there are 4 Jacks and 52 cards in the deck. Also, the probability of drawing a Queen and King is 4/52. So, if you draw one card from a normal deck of cards the probability of drawing a jack or queen or king is 4/52 + 4/52 + 4/52 = 12/52 or 3/13 or 0.2308.

Probability of getting the sum of 5 when rolling 2 dice is 4/36 or 1/9 or 0.111. There are 4 ways to get a 5: 2,3 & 3,2 & 1,4 & 4,1. There are 36 outcomes on a roll of two dice.

In detailed way.

The possible elements in the sample space is got by 62 = 36 So n(S) = 36

THe event A is defined as getting sum of 5. The possible elements are (1,4) (2,3) (3,2) (4,1). Thus n(A) = 4

Required prob. = n(A)/n(S) = 4/36 = 1/9

3 out of 8

1/7776

50/50, or one chance in each set of three rolls. No matter how many times it's rolled, every roll is one in six. Just like flipping a coin. it's 50/50 heads or tails, even if you got 'heads' 20 times in a row the next flip will still be 50/50 chance of 'heads' again

2, one jack of diamonds one jack of hearts.

The z-score is 0.84. In the related link, look in the body of the table for .3 area (.2995 closest) and it yields the .84 z-score.

Given any sample size there are many samples of that size that can be drawn from the population. In the population is N and the sample size in n, then there are NCn, but remember that the population can be infinite.

A test statistic is a value that is calculated from only the observations in a sample (no unknown parameters are estimated). The value of the test statistic will change from sample to sample. The sampling distribution of a test statistic is the probability distribution function for all the values that the test statistic can take across all possible samples.

The two dice sum of 4:

1,3

2,2

3,1

= 3 in 36

= 1 in 12

5 letters in first position there can be any one of the five, in the second position there are four possibilities left, in third position one of three remaining letters etc. 5 * 4 * 3 * 2 *1 = 120.

The probability that the sum is seven all three times is 1/216.

The probability of flipping a heads is 1/2 and the probability of rolling a 6 is 1/6. By the laws of probability it would be logical to multiply them together, (1/2)(1/6) thus the answer being 1/12 with is roughly eight percent.

If the pack is well shuffled, the probability is 1/52.

The probability is 21/26.