Algebra of Events
Let A and B are two events with sample space S. Then
A ∪ B is the event that either A or B or Both occur. (i.e., at least one of A or B occurs)
A ∩ B is the event that both A and B occur
A¯ is the event that A does not occur
A¯∩B¯ is the event that none of A and B occurs
Example : Consider a die is thrown , A be the event of getting 2 or 4 or 6 and B be the event of getting 4 or 5 or 6. Then
A = {2, 4, 6} and B = {4, 5, 6}
A ∪ B = {2, 4, 5, 6}
A ∩ B = {4, 6}
A¯ = {1, 3, 5}
B¯ = {1, 2, 3}
A¯∩B¯ = {1,3}
Probability of en Event
Let E be an event and S be the sample space. Then probability of the event E can be defined as
P(E) =n(E)/n(S)
where P(E) = Probability of the event E, n(E) = number of ways in which the event can occur and n(S) = Total number of outcomes possible
Examples
1) A coin is tossed once. What is the probability of getting Head?
Total number of outcomes possible when a coin is tossed = n(S) = 2 (∵ Head or Tail)
E = event of getting Head = {H}. Hence n(E) = 1
P(E) =n(E)/n(S)=1/2
Two dice are rolled. What is the probability that the sum on the top face of both the dice will be greater than 9?
Total number of outcomes possible when a die is rolled = 6 (∵ any one face out of the 6 faces)
Hence, total number of outcomes possible two dice are rolled, n(S) = 6 × 6 = 36
E = Getting a sum greater than 9 when the two dice are rolled = {(4, 6), {5, 5}, {5, 6}, {6, 4}, {6, 5}, (6, 6)}
Hence, n(E) = 6
P(E) =n(E)/n(S)=6/36=1/6
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