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A box contains cards numbered from 1 to 50. If one card is drawn at random, what is the probability that the number on the card is a multiple of 6 or 9?
0.2
0.22
0.24
0.26
0.22
Count multiples of 6 (50/6 = 8) and 9 (50/9 = 5). Subtract multiples of the LCM of 6 and 9 (18), which are 2 (18 and 36). Result is (8 + 5 - 2)/50 = 11/50 = 0.22.
Total cards range from 1 to 50. Events are multiples of 6 and 9.
P(AтИкB)=P(A)+P(B)тИТP(AтИйB)
Count multiples of 6 (50/6 = 8) and 9 (50/9 = 5). Subtract multiples of the LCM of 6 and 9 (18), which are 2 (18 and 36). Result is (8 + 5 - 2)/50 = 11/50 = 0.22.
Students often forget to subtract the intersection (multiples of 18) and double-count numbers like 18 and 36.
Identify multiples of 6
Find count of multiples of 6 up to 50 by calculating floor division 50├╖6=8.33, giving 8 multiples.
n(A)=тМК650тАЛтМЛ=8
Identify multiples of 9
Find count of multiples of 9 up to 50 by calculating floor division 50├╖9=5.55, giving 5 multiples.
n(B)=тМК950тАЛтМЛ=5
Identify multiples of both (LCM)
Find multiples of LCM(6,9)=18 up to 50, which are 18 and 36, giving 2 multiples.
n(AтИйB)=тМК1850тАЛтМЛ=2
Apply inclusion-exclusion principle
Calculate the number of favorable outcomes: 8+5тИТ2=11. Then divide by total outcomes (50).
P(AтИкB)=508+5тИТ2тАЛ=5011тАЛ=0.22
B is correct because the number of favorable outcomes is 11 out of 50 total cards, resulting in a probability of 0.22.
This concept of overlapping sets is identical to finding the number of elements in the union of two sets in Arithmetic Progressions and Number Theory problems.