1. Counting theory:
We may all remember: Given n objects, the number of ways of ordering these objects is:
n! = n(n-1)(n-2)..3.2.1.
n choose k, which is the number of distinct ways of choosing k objects from n: (n k) = n! / ( k! * (n-k)! )
For example:
If we want to form a random group of 3 students among 20 students, there are:
20! / (3! * 17!) = 1140 possible groups
2. Probability
- First problem: Discrete probability distribution for the sum of two dice.
- Second problem: Two people take turns trying to sink a basketball into a net. Person 1 succeeds with probability 1/3, person 2 with 1/4. What is the probability that person 1 succeeds before person 2?
We may all remember: Given n objects, the number of ways of ordering these objects is:
n! = n(n-1)(n-2)..3.2.1.
n choose k, which is the number of distinct ways of choosing k objects from n: (n k) = n! / ( k! * (n-k)! )
For example:
If we want to form a random group of 3 students among 20 students, there are:
20! / (3! * 17!) = 1140 possible groups
2. Probability
- First problem: Discrete probability distribution for the sum of two dice.
- Second problem: Two people take turns trying to sink a basketball into a net. Person 1 succeeds with probability 1/3, person 2 with 1/4. What is the probability that person 1 succeeds before person 2?
Solution for problem 2:
ReplyDeleteLet E be the event that person 1 succeeds before person 2.
Aj be the event that person 1 succeeds before person 2 and the first success is on trial number j.
Hence, P(E) = P(A1) + P(A2) + .. + P(A-infinite)
P(A1): person 1 succeeds on the first trial = 1/3
P(A2): person 1 misses, 2 misses, then 1 succeeds = 2/3 * 3/4 * 1/3 = 1/2 * 1/3
.
.
.
P(Aj) = (1/2)^(j-1) * (1/3)
Therefore,
P(E) = 2/3
Note:
We used: sum_(j=k)^{infinite}{r^j} = r^k/(1-r)