# Determining the Number of members in a sample space

Determination of the number of members of an event and the number of members of a sample space, we can not always list first. Sometimes in determining it, we need to use rule of enumerating as that discussed in the beginning.
Example 4:
Inside a box there are 8 red marbles and 4 white marbles. Three marbles will be taken at the same time. Determine the probability of the taken marbles, If:
a. Three red marbles are taken.
b. Two red marbles and a white one are taken.
The number of marbles in the box =8+4=12.
Three marbles will be taken at the same time.
The number of ways to take three from the twelve marbles is equal to

a. The number of ways to take 3 red marbles from the 8 red marbles is equal to

b. The number of ways to take 2 red marbles and a white one from the 8 red marbles is equal to

The number of ways to take 1 white marble and from the 4 white marbles which are provided

The number of ways to take 2 red marbles from the 8 which are provided and to take 1 white marble from the 4 which are provided =28∙4=112. The probability when the 2 taken is red and another is white =112/220.

Exercise Competency test 4
1. There are 7 red marbles and 3 blue marbles in a box. The probability of the 3 taken marbles as the same time are red =…
A. 3/10 B. ⅓ C. 7/24 D. ¼ E. 3/7
That means 3 red marbles and none blue marbles are taken. This event is denoted by K. Use combination to enumerate the expected event (K) and members of the sample space (S) because phrase “same time” here shows there is no sequence.

2. A coin is tossed 5 times. The probability of three times of number side appearing is equal to …
A. ⅗ B. ½ C. ⅖ D. 5/16 E. ⅕
A coin has two sides, suppose we call the number side (N) for the first side and we call the image side (E) for the second side. Members of the expected event occuring are K={NNNEE,…}→n(N)=3;n(E)=2. Remember permutation with repetition items.

The number of members of the sample space is n(S)=25=32.

Another way, Use the probability distribution formula:

Generally:
p is the probability of expected event occuring,
q is the probability of expected event does not occur,
n is the number of same treatment,
x is the expectation frequency,
Here, p is the probability of number side of a coin appearing and q is the probability of image side of a coin appearing.

3. A set of bridge card contains 52 cards, among there are 13 heart cards. If 4 cards as taken at the same time, the probability of the 3 taken cards are heart =…