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A die is rolled. What is the probability that the number shown is a prime number or an even number?
5/6
1/2
2/3
1/3
5/6
Identify the set of favorable outcomes: {1, 2, 3, 4, 5, 6}. Prime numbers are {2, 3, 5} and even numbers are {2, 4, 6}. Combining these gives {2, 3, 4, 5, 6}, which contains 5 elements out of 6.
A single fair six-sided die is rolled with sample space S = {1, 2, 3, 4, 5, 6}.
P(AтИкB)=P(A)+P(B)тИТP(AтИйB)
Identify the set of favorable outcomes: {1, 2, 3, 4, 5, 6}. Prime numbers are {2, 3, 5} and even numbers are {2, 4, 6}. Combining these gives {2, 3, 4, 5, 6}, which contains 5 elements out of 6.
Students often count 2 twice (as both prime and even) and incorrectly get 6/6, forgetting that in probability, intersection elements must be subtracted once.
Define Sample Space and Events
The sample space is S={1,2,3,4,5,6} with тИгSтИг=6. Let event A be prime numbers {2,3,5} and event B be even numbers {2,4,6}.
тИгSтИг=6,A={2,3,5},B={2,4,6}
Identify Intersection
Identify common elements in both sets. The number 2 is both prime and even, so AтИйB={2}.
AтИйB={2},тИгAтИйBтИг=1
Apply Addition Rule
Apply the inclusion-exclusion principle: тИгAтИкBтИг=тИгAтИг+тИгBтИгтИТтИгAтИйBтИг. Here, тИгAтИг=3 and тИгBтИг=3.
тИгAтИкBтИг=3+3тИТ1=5
Calculate Final Probability
Divide the number of favorable outcomes by the total outcomes in the sample space.
P(AтИкB)=65тАЛ
A is correct because the set of prime or even numbers is {2, 3, 4, 5, 6}, resulting in 5 favorable outcomes out of 6 total possibilities.
This concept of union of sets is fundamental in Permutation & Combination as well as Venn diagram-based Data Interpretation questions.