Re: George camed 80, 85, and 90 on the first three tests in his geography
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23 Jun 2024, 12:06
We need to determine the probability that George's final average score is more than 85. Remember the basic probability formula:
Probability = (desired outcomes / possible outcomes).
We will start by calculating the total number of desired outcomes.
Then we will calculate the total number of possible outcomes.
The total number of desired outcomes equals the number of ways that George can get a final average score greater than 85.
The total number of possible outcomes equals the number of ways that George can get any possible score.
Now, we know that George has already done three tests. And we know that the maximum score on each test is 100. So, we start by figuring out the minimum score George needs on the final test to get an average score greater than 85 across all four tests.
We use a simple formula for averages:
Average score = (total score / number of tests)
But, the average has to be greater than 85, so we turn the equation into an inequality:
85 < (total score / 4)
This can be rewritten as:
85*4 < total score
Or,
340 < total score
Next, George's scores on the first three tests are 80, 85, and 90. The total score George has already achieved is 80 + 85 + 90 = 255
Therefore, the score George needs on the final test to exceed a total of 340 points:
340 < 255 + final score
340 - 255 < final score
85 < final score
Therefore, George needs to score more than 85 on the final test, which means he needs at least 90 points (since scores are in multiples of 10). So, out of 10 questions, George must answer either 9 or 10 questions correctly.
But wait! You might be tempted to think this gives us two desired outcomes. But that would be a mistake!! We must consider the number of ways George can answer 9 or 10 questions correctly to get the total number of possible outcomes. Let me illustrate this:
George answers 9 questions correctly:
Scenario 1) YYYNYYYYYY
Scenario 2) NYYYYYYYYY
Are scenario 1 & 2 the same? No! So, in how many ways can George answer 9 questions correctly? We will use the permutation formula (nPr):
nPr = (n!) / (n-r)!
Where n is the total number of objects and r is the number of selected objects.
Plugging in the values we get:
10! / (10-1)!
Or,
10! / (9)!
Which simplifies to:
10/1 or 10.
And, in how many ways can George answer all questions correctly? Only one. Let me illustrate this:
Scenario 1) YYYYYYYYYY
Scenario 2) YYYYYYYYYY
Are these scenarios the same? Yes! So, there is only one way in which George can answer all 10 questions correctly.
Therefore, the total number of desired outcomes is 10 + 1 = 11.
Next we will calculate the total number of possible outcomes. George can answer any number of questions correctly. So, we will use the combination formula nCr = (n!) / r!(n-r)! to calculate the total number of ways George can answer zero, one, two, three, four, five, six, seven, eight, nine and ten questions correctly. This will be a time consuming and cumbersome process…
George answers zero questions correctly:
Do not use the combinations formula for this! The answer is simply 1.
George answers one question correctly:
nCr = 10! / 1! (10-1)! = 10
George answers two questions correctly:
nCr = 10! / 2! (10-2)! = 45
George answers three questions correctly:
nCr = 10! / 3! (10-3)! = 120
George answers four questions correctly:
nCr = 10! / 4! (10-4)! = 210
George answers five questions correctly:
nCr = 10! / 5! (10-5)! = 252
George answers six questions correctly:
nCr = 10! / 6! (10-6)! = 210
George answers seven questions correctly:
nCr = 10! / 7! (10-7)! = 120
George answers eight questions correctly:
nCr = 10! / 8! (10-8)! = 45
George answers nine questions correctly:
nCr = 10! / 9! (10-9)! = 10
George answers ten questions correctly:
Do not use the combinations formula for this! The answer is simply 1.
Then you need to ADD these combinations to find the total number of ways George could get any possible score on the final test (George could answer two questions correctly OR three OR four etc).
1+ 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 = 1024
Therefore the probability is:
11/1024 = Answer choice D.