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A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is _____.
Correct answer is '214'. Can you explain this answer?
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A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let p ...
Solution:
Finding the total number of elements in the set
The set contains {1, 2, 3,…, 2000}.
Therefore, the number of elements in the set is 2000.
Finding the multiples of 3 in the set
The first multiple of 3 in the set is 3, and the last multiple of 3 is 1998.
Therefore, the number of multiples of 3 in the set is (1998-3)/3 + 1 = 666.
Finding the multiples of 7 in the set
The first multiple of 7 in the set is 7, and the last multiple of 7 is 1995.
Therefore, the number of multiples of 7 in the set is (1995-7)/7 + 1 = 286.
Finding the multiples of 21 in the set
The first multiple of 21 in the set is 21, and the last multiple of 21 is 1995.
Therefore, the number of multiples of 21 in the set is (1995-21)/21 + 1 = 94.
Finding the number of elements that are multiples of 3 or 7
The number of elements that are multiples of 3 or 7 is equal to the number of multiples of 3 plus the number of multiples of 7 minus the number of multiples of 21.
Therefore, the number of elements that are multiples of 3 or 7 is 666 + 286 - 94 = 858.
Finding the probability that a number chosen at random is a multiple of 3 or 7
The probability that a number chosen at random is a multiple of 3 or 7 is equal to the number of elements that are multiples of 3 or 7 divided by the total number of elements in the set.
Therefore, the probability that a number chosen at random is a multiple of 3 or 7 is 858/2000 = 0.429.
Finding the value of 500p
To find the value of 500p, we need to multiply the probability by 500.
Therefore, 500p = 500 × 0.429 = 214.
Hence, the value of 500p is 214.
Finding the total number of elements in the set
The set contains {1, 2, 3,…, 2000}.
Therefore, the number of elements in the set is 2000.
Finding the multiples of 3 in the set
The first multiple of 3 in the set is 3, and the last multiple of 3 is 1998.
Therefore, the number of multiples of 3 in the set is (1998-3)/3 + 1 = 666.
Finding the multiples of 7 in the set
The first multiple of 7 in the set is 7, and the last multiple of 7 is 1995.
Therefore, the number of multiples of 7 in the set is (1995-7)/7 + 1 = 286.
Finding the multiples of 21 in the set
The first multiple of 21 in the set is 21, and the last multiple of 21 is 1995.
Therefore, the number of multiples of 21 in the set is (1995-21)/21 + 1 = 94.
Finding the number of elements that are multiples of 3 or 7
The number of elements that are multiples of 3 or 7 is equal to the number of multiples of 3 plus the number of multiples of 7 minus the number of multiples of 21.
Therefore, the number of elements that are multiples of 3 or 7 is 666 + 286 - 94 = 858.
Finding the probability that a number chosen at random is a multiple of 3 or 7
The probability that a number chosen at random is a multiple of 3 or 7 is equal to the number of elements that are multiples of 3 or 7 divided by the total number of elements in the set.
Therefore, the probability that a number chosen at random is a multiple of 3 or 7 is 858/2000 = 0.429.
Finding the value of 500p
To find the value of 500p, we need to multiply the probability by 500.
Therefore, 500p = 500 × 0.429 = 214.
Hence, the value of 500p is 214.
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Community Answer
A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let p ...
Let A = {1, 2, 3 ...., 2000}
Let E1 = 3m, 1 ≤ m ≤ 666, m ∈ N
E2 = 7k, 1 ≤ k ≤ 285, k ∈ N
= n(E1 ∪ E2) = 666 + 285 - 95 = 856
= P(E1 ∪ E2) = 856/2000 = 500β = 856/4 = 214
Let E1 = 3m, 1 ≤ m ≤ 666, m ∈ N
E2 = 7k, 1 ≤ k ≤ 285, k ∈ N
= n(E1 ∪ E2) = 666 + 285 - 95 = 856
= P(E1 ∪ E2) = 856/2000 = 500β = 856/4 = 214
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A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is _____.Correct answer is '214'. Can you explain this answer?
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