PROBABILITY OF A OR B
4.4 PROBABILITY OF A OR B
If events A and B are mutually exclusive (Two events are mutually exclusive if it is not possible for both of them to occur at the same time. For example, if a die is rolled, the event "getting a 1" and the event "getting a 2" are mutually exclusive since it is not possible for the die to be both a one and a two on the same roll. The occurrence of one event "excludes" the possibility of the other event.), then the probability of A or B is simply:
P (A or B) = p (A) + p (B).
What is the probability of rolling a die and getting either a 1 or a 6? Since it is impossible to get both a 1 and a 6, these two events are mutually exclusive. Therefore,
P (1 or 6) = p (1) + p (6) = 1/6 + 1/6 = 1/3
If the events A and B are not mutually exclusive, then
P (A or B) = p (A) + p (B) - p (A and B).
The logic behind this formula is that when p (A) and p (B) are added, the occasions on which
A and B both occur are counted twice. To adjust for this, p (A and B) is subtracted. What is the probability that a card selected from a deck will be either an ace or a spade? The relevant probabilities are:
P (ace) = 4/52 p (spade) = 13/52
The only way in which an ace and a spade can both be drawn is to draw the ace of spades. There is only one ace of spades, so:
p (ace and spade) = 1/52.
The probability of an ace or a spade can be computed as:
p (ace) +p (spade)-p (ace and spade) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13.
Consider the probability of rolling a die twice and getting a 6 on at least one of the rolls. The events are defined in the following way:
Event A: 6 on the first roll: p(A) = 1/6 Event B: 6 on the second roll: p(B) = 1/6 p(A and B) = 1/6 x 1/6 p(A or B) = 1/6 + 1/6 - 1/6 x 1/6 = 11/36
The same answer can be computed using the following admittedly convoluted approach: Getting a 6 on either roll is the same thing as not getting a number from 1 to 5 on both rolls. This is equal to: 1 - p (1 to 5 on both rolls).
The probability of getting a number from 1 to 5 on the first roll is 5/6. Likewise, the probability of getting a number from 1 to 5 on the second roll is 5/6. Therefore, the probability of getting a number from 1 to 5 on both rolls is: 5/6 x 5/6 = 25/36. This means that the probability of not getting a 1 to 5 on both rolls (getting a 6 on at least one roll) is:
1-25/36 = 11/36.
Despite the convoluted nature of this method, it has the advantage of being easy to generalize to three or more events. For example, the probability of rolling a die three times and getting a six on at least one of the three rolls is:
1 - 5/6 x 5/6 x 5/6 = .421
Mutually exclusive: Two events are mutually exclusive if it is not possible for both of them to occur. For example, if a die is rolled, the event "getting a 1" and the event "getting a 2" are mutually exclusive since it is not possible for the die to be both a one and a two on the same roll. The occurrence of one event "excludes" the possibility of the other event.