MATH problems dealing with POSSIBILITIES. HELP IS APPRECIATED

Theo is participating in a free throw competition. to advance to the next round, he must make at least 12 of 15 free throws, If theo's free throw shooting percentage is 82%, what is the probability that he will advance to the next round?

Let p = 82% = 0.82, the probability that Theo will make any particular free-throw.

The number N of free-throws he makes out of 15 is a discrete random variable, with N taking on values n = {0, 1, 2, ..., 15}

So, the probability P(N = n) that Theo makes exactly n free-throws is given by:

P(N = n) = 15Cn × p^n × (1 - p)^(15 - n)

= 15Cn × 0.82^n × 0.18^(15 - n)

Theo will advance to the next round if he makes at least 12 free throws with probability P(N ≥ 12); that is, if he makes exactly 12, exactly 13, exactly 14 or exactly 15 free-throws. This gives us the sum probability:

P(N ≥ 12) = P(N = 12) + P(N = 13) + P(N = 14) + P(N = 15)

= 15C12 × 0.82^12 × 0.18^3
+ 15C13 × 0.82^13 × 0.18^2
+ 15C14 × 0.82^14 × 0.18^1
+ 15C15 × 0.82^15 × 0.18^0

≈ 0.7218050756

... or, about 72%

Let P(n) denote the probability that Theo makes exactly n shots. If Theo makes at least 12, that means that he made either 12, 13, 14, or 15. Therefore,

Prob[Theo makes at least 12] = P(12) + P(13) + P(14) + P(15).

Now we need only find a formula for P(n). Choose any n shots out of all 15. The probability that he makes exactly those n shots and misses all the others is (82%)^n * (18%)^{15-n}. There are C(15, n) ways that you could choose the n shots, where C denotes a binomial coefficient. Thus the probability that he makes exactly n shots is

P(n) = (82%)^n * (18%)^{15-n} * C(15, n).

Now you should be able to use this formula to determine the answer.

EDIT: The value Brandon calculated is correct. See the link below.

Set up a proportion. 82 over 100 = x over 15.theb it should be easier