Mathematical Research: Probability and Statistics

The probability of rolling a specific number on a die is calculated by dividing 1 by the total number of possible outcomes. In this case, there are six possible outcomes (1, 2, 3, 4, 5, 6), so the probability of rolling a 6 is 1/6.

The expected value of a random variable is calculated by multiplying each possible value by its probability and then summing the results. In this case, we have: E(X) = (1 * 0.2) + (2 * 0.5) + (3 * 0.3) = 0.2 + 1.0 + 0.9 = 2.0.

The standard deviation is a measure of the spread or variability of a dataset. It is calculated by taking the square root of the variance, which is the average of the squared differences between each data point and the mean.

There are 13 hearts in a standard deck of 52 playing cards. Therefore, the probability of drawing a heart is 13/52, which simplifies to 1/13.

There are four possible outcomes when flipping a coin twice: HH, HT, TH, TT. Only one of these outcomes (TT) does not have at least one head. Therefore, the probability of getting at least one head is 3/4.

There are 36 possible outcomes when rolling two six-sided dice. The outcomes that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Therefore, the probability of getting a sum of 7 is 6/36, which simplifies to 1/6.

There are 36 possible outcomes when rolling two six-sided dice. The outcomes that sum to 10 or less are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (6, 1). Therefore, the probability of getting a sum of 10 or less is 21/36, which simplifies to 5/6.

There are 36 possible outcomes when rolling two six-sided dice. The outcomes that sum to 12 or more are (6, 6), (5, 6), (6, 5), (4, 6), (6, 4), (3, 6), (6, 3), (2, 6), (6, 2). Therefore, the probability of getting a sum of 12 or more is 9/36, which simplifies to 1/4.

There are 36 possible outcomes when rolling two six-sided dice. The outcomes that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). The outcomes that sum to 11 are (5, 6), (6, 5). Therefore, the probability of getting a sum of 7 or 11 is (6 + 2)/36 = 8/36, which simplifies to 1/6.

There are 36 possible outcomes when rolling two six-sided dice. The outcomes that sum to 2 are (1, 1). The outcomes that sum to 3 are (1, 2), (2, 1). The outcomes that sum to 12 are (6, 6). Therefore, the probability of getting a sum of 2, 3, or 12 is (1 + 2 + 1)/36 = 4/36, which simplifies to 1/9.

There are 36 possible outcomes when rolling two six-sided dice. The outcomes that sum to 4 are (1, 3), (2, 2), (3, 1). The outcomes that sum to 5 are (1, 4), (2, 3), (3, 2), (4, 1). The outcomes that sum to 6 are (1, 5), (2, 4), (3, 3), (4, 2), (5, 1). The outcomes that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). The outcomes that sum to 8 are (2, 6), (3, 5), (4, 4), (5, 3), (6, 2). The outcomes that sum to 9 are (3, 6), (4, 5), (5, 4), (6, 3). The outcomes that sum to 10 are (4, 6), (5, 5), (6, 4). The outcomes that sum to 11 are (5, 6), (6, 5). Therefore, the probability of getting a sum of 4, 5, 6, 7, 8, 9, 10, or 11 is (3 + 4 + 5 + 6 + 5 + 4 + 3 + 2)/36 = 32/36, which simplifies to 5/6.

There are 36 possible outcomes when rolling two six-sided dice. The highest possible sum is 12. Therefore, the probability of getting a sum of 13 or more is 0.

There are 36 possible outcomes when rolling two six-sided dice. The outcomes that sum to 1 are (1, 1). The outcomes that sum to 13 are (6, 6). Therefore, the probability of getting a sum of 1 or 13 is (1 + 1)/36 = 2/36, which simplifies to 1/18.

There are 36 possible outcomes when rolling two six-sided dice. The outcomes that sum to 2 are (1, 1). The outcomes that sum to 12 are (6, 6). Therefore, the probability of getting a sum of 2 or 12 is (1 + 1)/36 = 2/36, which simplifies to 1/18.