Three cards are drawn all at once from a standard deck of 52 playing cards. what is the probability of selecting exactly two aces?

There are 4 aces in a standard 52 card deck. So the probability of getting an ace on the first try is 4/52, or 1/13 (simplified). The probability of getting it the second time after you pulled one out is 3/51 because you removed one, or 1/17 (simplified). The probability of NOT getting a ace on the third go is 48/50, because there are 50 remaining cards and 2 of them remain to be aces. Simplified probability of not getting it is 24/25. Multiply these probabilities together to find the exact probability of getting exactly two aces.

1/13 * 1/17 * 24/25 = 24/5525 or 0.0043438914027149.

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x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.