I am trying to learn probability, and having a hard time to be honest. What I try to do is to break a problem in terms of event space. For eg, event space for rolling a die is {1,2,3,4,5,6}. For rolling two dice it would be 36 elements with pairs from 1,1 to 6,6. Then I try to calculate probability of a given event using simple basic formula of occurrences/total events.
On topic of probability of independent events, suppose I roll 2 dice and want to calculate probability of 2 on first die and 5 on second die. Book I am reading uses "And" rule for independent events:
P(2 and 5) = P(2) * P(5)
I am not sure of this approach. When we say event "2 in first roll, 5 in second roll", we basically are talking of an event "2 rolls of die". And so our event space consists of 36 elements, each representing a pair as discussed above. But in above formula, we are multiplying probabilities of events which come from a different event space(containing 6 elements). So basically we are trying to solve problem belonging to some event space using probabilities from different event space.
Am I wrong in my thinking here?
Best Answer
Honestly the simple way is to draw the sample space, that is the following
and immediately you have a clear sight of the elements you need to calculate your probability.
Example 1: sum of the two dice $\leq 7$ and first die =1
Result: [$\frac{6}{36}$]
Example 2: first die 2 and second die 5, as you requested: result $\frac{1}{36}$
Example 3: sum of the two dice $\leq 7$ and at least one "1": Result $\frac{11}{36}$
and so on...any question you have to answer you have only to "visualize" and count the favourable events in the above table and divide it by $36$
PS: the plural of "die" is "dice", I amended you post