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A bag contains cards numbered 1 to 50. One card is drawn at random. What is the probability that the number is a multiple of 5 or 10?
1/5
1/10
1/2
3/10
1/5
Since every multiple of 10 is automatically a multiple of 5, the set of multiples of 10 is a subset of multiples of 5. Thus, we only need to count the multiples of 5.
Total cards in a bag range from 1 to 50. We need to find the probability of drawing a card that is a multiple of 5 or 10.
P(AтИкB)=P(A)+P(B)тИТP(AтИйB)
Since every multiple of 10 is automatically a multiple of 5, the set of multiples of 10 is a subset of multiples of 5. Thus, we only need to count the multiples of 5.
Students often double-count multiples of 10 by adding the count of multiples of 5 to the count of multiples of 10, failing to realize that multiples of 10 are already contained within the multiples of 5.
Define the sample space
The total number of cards is n(S)=50.
n(S)=50
Identify multiples of 5
Multiples of 5 are 5,10,15,тАж,50. The number of such terms is given by 550тАЛ=10.
n(Multiple┬аof┬а5)=10
Identify multiples of 10
Multiples of 10 are 10,20,30,40,50. The number of such terms is 1050тАЛ=5. Since all these are already included in the multiples of 5, the union remains the set of multiples of 5.
n(Multiple┬аof┬а5тИкMultiple┬аof┬а10)=10
Calculate the probability
Applying the probability formula P=Total┬аOutcomesFavorable┬аOutcomesтАЛ, we get P=5010тАЛ.
P=5010тАЛ=51тАЛ
A is correct because the set of multiples of 10 is a subset of the multiples of 5, resulting in 10 favorable outcomes out of 50, which simplifies to 1/5.
This concept of overlapping sets is identical to finding the union of two events in Venn Diagrams, frequently used in both Probability and Logical Reasoning sections.