William has a pair of identical number cubes. The faces of each cube are numbered 1 through 6. William will roll the cubes one time. What is the probability that the numbers showing face-up after the roll will have a sum of 9?
A. Start Fraction 1 over 18 End Fraction
B. start fraction 1 over 9 end fraction
C. three-fourths
D. start fraction 8 over 9 end fraction
There are several ways to obtain a sum of 9 when rolling two dice, including: (3,6), (4,5), (5,4), and (6,3). Each of these outcomes has a probability of $\frac16 \cdot \frac16 = \frac{1}{36}$ of occurring, since there are 6 possible outcomes for each die and they are assumed to be fair and independent.
Therefore, the probability of obtaining a sum of 9 is the sum of the probabilities of these four outcomes: $\frac{1}{36} + \frac{1}{36} + \frac{1}{36} + \frac{1}{36} = \frac{4}{36} = \frac{1}{9}$.
The answer is $\boxed{\textbf{(B) } \frac{1}{9}}$.
How many different arrangements can be made with the letters from the word MATH?
A. 12
B. 24
C. 106
D. 1,210
There are four letters in the word MATH, so there are 4 choices for the first letter, 3 choices for the second letter (since one has already been chosen), 2 choices for the third letter, and only 1 choice for the last letter.
Therefore, the total number of arrangements is $4 \cdot 3 \cdot 2 \cdot 1 = \boxed{\textbf{(B) }24}$.
To find the probability of rolling a sum of 9 on two identical number cubes, we need to determine the number of favorable outcomes (rolling a sum of 9) and divide it by the total number of possible outcomes.
To start, let's determine the number of favorable outcomes. The sum of two numbers on the two cubes needs to be 9. We can make a list of all possible outcomes:
(3, 6),
(4, 5),
(5, 4),
(6, 3)
So, there are 4 possible favorable outcomes.
Next, let's determine the total number of possible outcomes. Each cube has 6 sides, so each cube can land on any of the 6 numbers. Since there are two cubes, the total number of outcomes is 6 * 6 = 36.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:
Probability = Number of favorable outcomes / Total number of outcomes
= 4 / 36
= 1/9
Therefore, the probability of rolling a sum of 9 is 1/9.
Hence, the answer is B. start fraction 1 over 9 end fraction.