Question

A group of 30 students from your school is part of the audience for a TV game show. The total number of people in the audience is 140. What is the theoretical probability of 4 students from your school being selected as contestants out of 8 possible contestant spots??

Use combination formula

Answer 1

Are you sure

Answer 2

Are you sure that's correct now

Answer 3

The combination formula is used to calculate the number of ways to choose a certain number of items from a larger set.

The formula for combinations is:

C(n, k) = n! / (k! * (n - k)!)

Where:
- n is the total number of items
- k is the number of items to choose

In this case, we want to calculate the number of ways to choose 4 students out of the 30 from your school. So n = 30 and k = 4.

C(30, 4) = 30! / (4! * (30 - 4)!)
C(30, 4) = 30! / (4! * 26!)

Now, calculate the factorials:

30! = 30 x 29 x 28 x 27 x 26!
4! = 4 x 3 x 2 x 1 = 24

Now plug these values back into the formula:

C(30, 4) = (30 x 29 x 28 x 27 x 26!) / (24 x 26!)
C(30, 4) = (30 x 29 x 28 x 27) / 24
C(30, 4) = 657720

Thus, there are 657720 ways to choose 4 students out of the 30 from your school.

Now, out of the 140 people in the audience, 8 are chosen as contestants. So the total number of ways to choose 8 contestants out of 140 is calculated in a similar way:

C(140, 8) = 140! / (8! * (140 - 8)!)

Now, calculate the factorials:

140! = 140 x 139 x 138 x ... x 133 x 132!
8! = 8 x 7 x 6 x ... x 2 x 1 = 40320

Now plug these values back into the formula:

C(140, 8) = (140 x 139 x 138 x ... x 132!) / 40320
C(140, 8) = 122728115896920

Therefore, the theoretical probability of 4 students from your school being selected as contestants out of 8 possible contestant spots is 657720 / 122728115896920 = 0.00000535.

Answer 4

I apologize for the error in my previous response. Let me correct that calculation.

To find the theoretical probability of 4 students from your school being selected as contestants out of 8 possible spots, we first need to calculate the total number of ways to choose 4 students from the 30 in your school and the total number of ways to choose 4 students from the 140 in the audience.

The number of ways to choose 4 students from the 30 in your school is given by the combination formula:

C(30, 4) = 30! / (4! * (30 - 4)!)
C(30, 4) = 30! / (4! * 26!)
C(30, 4) = (30*29*28*27) / (4*3*2*1)
C(30, 4) = 27,405

The number of ways to choose 4 students from the 140 in the audience is given by:

C(140, 4) = 140! / (4! * (140 - 4)!)
C(140, 4) = 140! / (4! * 136!)
C(140, 4) = (140*139*138*137) / (4*3*2*1)
C(140, 4) = 703,698,840

Therefore, the theoretical probability of 4 students from your school being selected as contestants out of 8 possible contestant spots is given by the ratio of these two values:

P = (27,405 / 703,698,840)
P ≈ 0.0000389

So, the correct theoretical probability is approximately 0.0000389. Thank you for pointing out the error, and I appreciate your attention to detail.

Answer 5

Let's recalculate the theoretical probability of 4 students from your school being selected as contestants out of 8 possible contestant spots.

The number of ways to choose 4 students from the 30 in your school is given by the combination formula:

C(30, 4) = 30! / (4! * (30 - 4)!)
C(30, 4) = 30! / (4! * 26!)
C(30, 4) = (30 * 29 * 28 * 27) / (4 * 3 * 2 * 1)
C(30, 4) = 27,405

The number of ways to choose the remaining 4 students from the 140 - 30 = 110 students who are not from your school is given by the combination formula:

C(110, 4) = 110! / (4! * (110 - 4)!)
C(110, 4) = 110! / (4! * 106!)
C(110, 4) = (110 * 109 * 108 * 107) / (4 * 3 * 2 * 1)
C(110, 4) = 1,645,780

Now that we have the number of ways to choose 4 students from your school and 4 students from the rest of the audience, we can calculate the total number of ways to choose 8 contestants out of the 140 in the audience:

C(140, 8) = 140! / (8! * (140 - 8)!)
C(140, 8) = 140! / (8! * 132!)
C(140, 8) = (140 * 139 * ... * 133 * 132!) / (8 * 7 * ... * 2 * 1)
C(140, 8) = 2,344,077,120

Therefore, the total number of ways for 4 students from your school to be selected as contestants out of 8 spots is:

27,405 * 1,645,780 = 45,067,058,900

The theoretical probability is the ratio of the number of favorable outcomes to the total number of outcomes:

P = 45,067,058,900 / 2,344,077,120
P ≈ 0.0192

So, the correct theoretical probability of 4 students from your school being selected as contestants out of 8 possible spots is approximately 0.0192. Thank you for your patience, and I apologize for the previous errors.