Question

At a local high school, 95 students have permission to park on campus. Each month, the student council holds a 'golden ticket parking lottery.' The three lucky winners are given reserved parking spots next to the main entrance. Last month, when all three golden tickets went to members of the same class, some people thought the lottery had been rigged. There are 28 students in the class, all of whom are eligible to park on campus.

a. Find the number of ways to choose any 3 students in the school as the 3 golden ticket winners in the lottery.
b. Find the number of ways to choose 3 students from the same class of 28 students.
c. Find the probability that the lottery would result in all three golden tickets going to students in the same class.
d. Based on your answer to part (c), do you think the lottery was carried out fairly?

Answer 1

a. There are 456,460 ways to choose any 3 students in the school as the golden ticket winners in the lottery. b. There are 3,276 ways to choose 3 students from the same class of 28 students as the golden ticket winners. c. The probability of all three golden tickets going to students in the same class is approximately 0.00718 or 0.718%.

Step-by-step explanation:

a. The number of ways to choose any 3 students in the school as the 3 golden ticket winners in the lottery is determined by the combination formula. The formula for combinations is nCr = n! / (r! * (n-r)!), where n is the total number of students in the school (95 in this case) and r is the number of students to be chosen (3 in this case). Using this formula, the number of ways to choose 3 students from 95 is:

nCr = 95! / (3! * (95-3)!) = 95! / (3! * 92!) = (95 * 94 * 93) / (3 * 2 * 1) = 456,460

Therefore, there are 456,460 ways to choose any 3 students in the school as the golden ticket winners in the lottery.

b. To find the number of ways to choose 3 students from the same class of 28 students, we can use the combination formula again. The formula remains the same, but now n is equal to 28 and r is equal to 3:

nCr = 28! / (3! * (28-3)!) = 28! / (3! * 25!) = (28 * 27 * 26) / (3 * 2 * 1) = 3,276

Therefore, there are 3,276 ways to choose 3 students from the same class of 28 students as the golden ticket winners in the lottery.

c. The probability that the lottery would result in all three golden tickets going to students in the same class can be found by dividing the number of ways to choose 3 students from the same class (3,276) by the number of ways to choose any 3 students in the school (456,460):

Probability = 3,276 / 456,460 = 0.00718

Therefore, the probability of all three golden tickets going to students in the same class is approximately 0.00718 or 0.718%.

d. Based on the probability calculated in part (c), it does not seem likely that the lottery was carried out fairly. The probability of all three golden tickets going to students in the same class is quite low (0.718%), suggesting that the outcome may have been influenced by external factors.

Answer 2

There are 12,650 ways to choose any 3 students as the golden ticket winners. There are 3,276 ways to choose 3 students from the same class. The probability of all three golden tickets going to students in the same class is approximately 25.90%.

Step-by-step explanation:

In order to answer the questions, we need to use the concept of combinations since we are selecting a set of students without regard to their arrangement.

a. To find the number of ways to choose any 3 students in the school as the 3 golden ticket winners in the lottery, we use the formula for combinations: nCr = n! / [(n - r)! * r!]. In this case, there are 95 students eligible, so there are 95C3 = 95! / [(95 - 3)! * 3!] = 12,650 ways to choose the winners.

b. To find the number of ways to choose 3 students from the same class of 28 students, we use the same formula for combinations. In this case, there are 28C3 = 28! / [(28 - 3)! * 3!] = 3,276 ways to choose the winners.

c. To find the probability that the lottery would result in all three golden tickets going to students in the same class, we divide the number of ways to choose 3 students from the same class (3,276) by the total number of ways to choose any 3 students (12,650). So the probability is 3,276 / 12,650 ≈ 0.2590, or approximately 25.90%.

d. Based on the answer to part (c), we can see that the probability is less than 50%. Therefore, it is unlikely that the lottery was rigged. The fact that all three golden tickets went to students in the same class can be attributed to chance rather than unfair manipulation of the results.