Elliott has eight numbered cards. One of the cards is chosen at random. Elliott says: * The probability of an 8 is [tex]$\frac{1}{4}$[/tex] * The range of the numbers is 5. * The probability of a number greater than 10 is 0. * The probability of a 7 is [tex]$\frac{1}{2}$[/tex] Fill in the six missing numbers.
Answer (2)
Determine the number of cards with 8: 4 1 × 8 = 2 .
Determine the number of cards with 7: 2 1 × 8 = 4 .
Use the range of 5 to determine the smallest number: 8 − x = 5 , so x = 3 .
The six missing numbers are 3, 7, 7, 7, 7, 8. Therefore, the numbers on the cards are: 3 , 6 , 7 , 7 , 7 , 7 , 8 , 8 .
Explanation
Analyze the problem and given data We are given that Elliott has eight numbered cards, with two of them being 6 and 8. We are also given the following probabilities:
P(8) = 4 1
Range = 5
P(number > 10) = 0
P(7) = 2 1
Our goal is to find the six missing numbers on the cards.
Determine the number of cards with 7 and 8 Since P(8) = 4 1 , this means that 4 1 of the cards have the number 8. Since there are 8 cards in total, the number of cards with 8 is 4 1 × 8 = 2 . This matches the information given in the problem.
Since P(7) = 2 1 , this means that 2 1 of the cards have the number 7. So, the number of cards with 7 is 2 1 × 8 = 4 .
Use the range and probability information We know that P(number > 10) = 0, which means that none of the numbers on the cards are greater than 10. Therefore, all numbers are less than or equal to 10.
The range of the numbers is 5, which means that the difference between the largest and smallest number is 5.
Find the missing number We currently have the numbers 6, 8, 8, and four 7s. So the numbers are 6, 7, 7, 7, 7, 8, 8, x 8 . Let the smallest number be s and the largest number be l . We know that l − s = 5 . Since the largest number we have so far is 8, l must be less than or equal to 8. Therefore, s = l − 5 ≤ 8 − 5 = 3 . So the smallest number is at most 3.
Since we have 6, 7, and 8, the smallest number must be such that the range is 5. If the smallest number is x , then 8 − x = 5 , which means x = 3 . Therefore, the missing number is 3.
Verify the solution and state the missing numbers The numbers on the cards are 3, 6, 7, 7, 7, 7, 8, 8. Let's check if this satisfies the given conditions:
P(8) = 8 2 = 4 1 (Correct)
Range = 8 - 3 = 5 (Correct)
P(number > 10) = 0 (Correct)
P(7) = 8 4 = 2 1 (Correct)
Therefore, the six missing numbers are 3, 7, 7, 7, 7, 8.
Examples
This type of probability problem can be used in quality control. For example, a factory produces items, and some are defective. By randomly selecting items and checking if they are defective, we can estimate the probability of finding a defective item and the range of defects in the production line. This helps in maintaining the quality of the products.
In Elliott's set of eight cards, there are 2 cards showing 8, 4 cards showing 7, and the smallest number is 3, resulting in the complete set: 3, 6, 7, 7, 7, 7, 8, 8. The conditions about probabilities and range are satisfied with these numbers. Therefore, the final card numbers are 3, 6, 7, 7, 7, 7, 8, 8.
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