Two fair dice are rolled 3 times and the sums of the numbers that come up is recorded. Find the probability of these events. A) the sum 10 on each of the three rolls? B) the sum is 10 exactly twice in three rolls?

A) P(10 on three rolls) = 1/1728 = 0.0005787

B) P(10 in two of three rolls) = 11/1728 = 0.0063657

Step-by-step explanation:

Each die has 6 possible values, so a pair of dice have a total of 6*6=36 outcomes.

To find a sum of 10, the cases are:

(4,6), (5,5) and (6,4).

So to have a sum of 10 we have 3 cases among the 36 possible, therefore the probability is:

P(10) = 3/36 = 1/12

A)

If we want the sum of 10 in each of the three rolls, the probability is:

P(10 on three rolls) = P(10)^3 = (1/12)^3 = 1/1728 = 0.0005787

B)

If the sum is 10 in two of three rolls, in one roll we need the probability of the sum not being 10:

P(not 10) = 1 - P(10) = 1 - (1/12) = 11/12

So we have:

P(10 in two of three rolls) = P(10)^2 * P(not 10) = (1/12)^2 * (11/12)

P(10 in two of three rolls) = 11/1728 = 0.0063657

answer:  

step-by-step explanation:

here x represents the height of the pole,

and, the height of the shadow of the pole is 12 feet.

also, the angle of elevation from the top of the pole is 40°,

therefore,

by the low of sine,

which is the required equation for finding the value of x,

the arc length is found by finding the circumference of the full circle (2xpixradius) and then dividing by 4 to find just one quarter of the circle (90 degrees).