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3 years ago

Two dice are rolled. Use the complement rule to find that the probability that the sum of the dice is less than 10.

Mathematics

1 answer:

Katarina [22]3 years ago

6 0

Answer:

$\dfrac{5}{6}$

Step-by-step explanation:

When two dice are rolled, the sample space is

$\begin{array}{cccccc}(1,1)&(1,2)&(1,3)&(1,4)&(1,5)&(1,6)\\(2,1)&(2,2)&(2,3)&(2,4)&(2,5)&(2,6)\\(3,1)&(3,2)&(3,3)&(3,4)&(3,5)&(3,6)\\(4,1)&(4,2)&(4,3)&(4,4)&(4,5)&(4,6)\\(5,1)&(5,2)&(5,3)&(5,4)&(5,5)&(5,6)\\(6,1)&(6,2)&(6,3)&(6,4)&(6,5)&(6,6)\end{array}$

In total, 36 outcomes

First, find the probability that the sum is 10 or greater:

All favorable outcomes

10: $(4,6),\ (5,5),\ (6,4)$

11: $(5,6),\ (6,5)$

12: $(6,6)$

In total, 6 outcomes.

The probability that the sum is 10 or greater is

$P(A)=\dfrac{6}{36}=\dfrac{1}{6}$

Use the complement rule to find that the probability that the sum of the dice is less than 10:

$P(A^C)=1-P(A)=1-\dfrac{1}{6}=\dfrac{5}{6}$

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2 years ago

Find the area of the shaded region.

lbvjy [14]

Answer:

$48\pi\\\approx 150.796$

Step-by-step explanation:

1.Approach

To solve this problem, find the area of the larger circle, and the area of the smaller circle. Then subtract the area of the smaller circle from the larger circle to find the area of the shaded region.

2.Find the area of the larger circle

The formula to find the area of a circle is the following,

$A=(\pi)(r^2)$

Where (r) is the radius, the distance from the center of the circle to the circumference, the outer edge of the circle. ( $\pi$ ) represents the numerical constant (3.1415...). One is given that the radius of (8), substitute this into the formula and solve for the area,

$A_l=(\pi)(r^2)\\A_l=(\pi)(8^2)\\A_l=(\pi)(64)$

3.Find the area of the smaller circle

To find the area of the smaller circle, one must use a very similar technique. One is given the diameter, the distance from one end to the opposite end of a circle. Divide this by two to find the radius of the circle.

8 ÷2 = 4

Radius = 4

Substitute into the formula,

$A_s=(\pi)(r^2)\\A_s=(\pi)(4^2)\\A_s=(\pi)(16)$

4.Find the area of the shaded region

Subtract the area of the smaller circle from the area of the larger circle.

$A_l-A_s\\=64\pi - 16\pi\\=48\pi$

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