

3.1
Calculating the Probability of Several Events
You have learned by now how to calculate probability for one event occurring, for example, the probability of rolling a six on a die, or if you pull out a red ball from an urn.
Here you will learn how to calculate probability for several events and why we call it the product of probabilities.
There are two types of probabilities for several events, namely: 
 Where events are independent from each other, for example if you roll a die twice, the result of the second throw will not be affected by the first throw. For every throw, there are six different independent events from what the throw before showed.
 Where events are dependent upon each other. If we take for example the urn with balls, and imagine that you are calculating the probability of pulling out two red balls in a row. If you pull out a ball and then another one, without putting the first one back, the probability will change between the first and the second try.
We can try by showing calculations for both examples:

Independent from Each Other
Example A
You roll one die twice and are to calculate the probability of rolling two sixes in a row.
Try A
Number of favorable outcomes: 1
Total number of possible
outcomes: 6

Try B
Number of favorable outcomes: 1
Total number of possible
outcomes: 6

In order to calculate the probability for both of these events to occur, we take the product of both of them:
Total probability = 
1 
· 
1 
= 
1 
≈ 
0.027 = 2.7 % 



6 
6 
36 
Answer: The probability to roll two sixes in a row is about 2.7 %
Dependant on Each Other
Example B
You have an urn with 10 balls. 3 are red, 5 are blue, and 2 are black. What is the probability of pulling out two red balls without putting back the first ball?
Try A
Number of favorable outcomes: 3
Total number of possible
outcomes: 10

Try B
Number of favorable outcomes: 2
Total number of possible outcomes: 9

Here we can see that try B is dependent on try A. If you have already pulled out a red ball, then there are only two red balls left. The total number of balls before the second try is then nine.
We calculate the product of both of the probabilities:
Total probability = 
3 
· 
2 
= 
6 
≈ 
0.067 = 6.7 % 



10 
9 
90 
Answer: The probability that we pull out two red balls is about 6.7 %. 

