Probability

Definition:

Probability is the likelihood of something happening. When someone tells you the probability of something happening, they are telling you how likely that something is. When people buy lottery tickets, the probability of winning is usually stated, and sometimes, it can be something like 1/10,000,000 (or even worse). This tells you that it is not very likely that you will win.

Formula:

The formula for probability tells you how many choices you have over the number of possible combinations.

Example 1:

Suppose a coin is flipped 3 times. What is the probability of getting two tails and one head?

Solution: For this experiment, the sample space consists of 8 sample points.

S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}

Each sample point is equally likely to occur, so the probability of getting any particular sample point is 1/8. The event "getting two tails and one head" consists of the following subset of the sample space.

A = {TTH, THT, HTT}

The probability of Event A is the sum of the probabilities of the sample points in A. Therefore,

P(A) = 1/8 + 1/8 + 1/8 = 3/8

**In this case,you can use a tree diagram to look at which is getting two tails and one head faced coin.

Example 2:

Two dice are thrown together.
Use a tree diagram to find the probability that one number is even and the other is odd.

There are six possible scores on one die: {1, 2, 3, 4, 5, 6}
Of these, three are even: {2, 4, 6} and three are odd: {1, 3, 5}

    
                                      

So, the tree diagram looks like this:


                                        

 So the probability that one number is even and the other is odd


                                                             

Example 3:

Two dice are rolled, find the probability that the sum is
a) equal to 1
b) equal to 4
c) less than 13


Solution to Example 3:

The sample space S of two dice is shown below.

S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
         (2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
         (3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
         (4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
         (5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
         (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }  

a) Let E be the event "sum equal to 1". There are no outcomes which correspond to a sum equal to 1, hence

P(E) = n(E) / n(S) = 0 / 36 = 0 

b) Three possible outcomes give a sum equal to 4: E = {(1,3),(2,2),(3,1)}, hence

 P(E) = n(E) / n(S) = 3 / 36 = 1 / 12

c) All possible outcomes, E = S, give a sum less than 13, hence.

 P(E) = n(E) / n(S) = 36 / 36 = 1 


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