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Zorluk: ZorBasic Single-Event Probability

A box contains 90 tokens, numbered sequentially from 11 to 9090, inclusive. If one token is drawn at random from the box, what is the probability that the number on the token is a multiple of 44 or 66, but not a multiple of 88?

  1. A
    1345\frac{13}{45}
  2. B
    13\frac{1}{3}
  3. 1990\frac{19}{90}Cevap
  4. D
    29\frac{2}{9}
  5. E
    730\frac{7}{30}

Cevap

The probability that the selected number is a multiple of 4 or 6, but not a multiple of 8, is 1990\frac{19}{90}.
To find the probability, determine the number of favorable outcomes out of 90 total outcomes. The number of multiples of 4 is 22, and the number of multiples of 6 is 15. The numbers that are multiples of both 4 and 6 are multiples of 12, of which there are 7. By inclusion-exclusion, the number of integers that are multiples of 4 or 6 is 22+15−7=3022 + 15 - 7 = 30. Since all multiples of 8 are automatically multiples of 4, all 11 multiples of 8 in the range are included in these 30 numbers. Excluding the multiples of 8 leaves 30−11=1930 - 11 = 19 favorable integers. Thus, the probability is 1990\frac{19}{90}.

Adım Adım Çözüm

1
Find the count of multiples of 4 and multiples of 6 in the range from 1 to 90.
Multiples of 4: ⌊904⌋=22\lfloor \frac{90}{4} \rfloor = 22. Multiples of 6: ⌊906⌋=15\lfloor \frac{90}{6} \rfloor = 15.
Identify the size of each individual set of multiples.
2
Calculate the number of elements in the union of multiples of 4 or 6 using the Inclusion-Exclusion Principle.
Multiples of both 4 and 6 are multiples of lcm(4,6)=12\text{lcm}(4,6) = 12. Count of multiples of 12: ⌊9012⌋=7\lfloor \frac{90}{12} \rfloor = 7. Union size: 22+15−7=3022 + 15 - 7 = 30.
Avoid double-counting numbers that are divisible by both 4 and 6.
3
Identify and subtract the multiples of 8 within this union.
Multiples of 8 in the range: ⌊908⌋=11\lfloor \frac{90}{8} \rfloor = 11. Since every multiple of 8 is also a multiple of 4, all 11 multiples of 8 are contained within the union. Favorable outcomes: 30−11=1930 - 11 = 19.
Filter out numbers that satisfy the exclusion condition (multiples of 8).
4
Compute the single-event probability.
P=1990P = \frac{19}{90}.
Divide the number of favorable outcomes (19) by the total sample space (90).

Anahtar Kavram

Basic Single-Event Probability with Principle of Inclusion-Exclusion and Set Restriction
Tahmini Süre:2m 0s
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Basic Single-Event Probability

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