Question

The Senior Class sponsor at a high school has enough extra money in the Grad Night budget to pay for meals for 3 of the 9 volunteer faculty members chaperoning the students on the trip. The volunteers are comprised of 6 teachers and 3 paraeducators. The Senior Class Sponsor selects the 3 people who will receive free meal vouchers and states that the people were selected at random. The 3 people selected were all paraeducators. All of the volunteers were curious how no teachers were picked for a free meal voucher if the choice were truly random.

(a) Calculate the probability that randomly selecting 3 people from a group of 6 teachers and 3 paraeducators will result in selecting 3 paraeducators.
(b) Based on your answer to part (a), is there reason to doubt the Senior Class Sponsor’s claim that the 3 people were selected at random? Explain.
(c) An alternative to calculating the exact probability is to conduct a simulation to estimate the probability. A proposed simulation process is described below.
Each trial in the simulation consists of rolling three fair, six-sided dice, one die for each of the Grad Night chaperones. For each die, rolling a 1, 2, 3, or 4 represents selecting a teacher; rolling a 5 or 6 represents selecting a paraeducator. After 1,000 trials, the number of times the dice indicate selecting 3 paraeducators is recorded.
Does the proposed process correctly simulate the random selection of 3 paraeducators from a group of 9 people consisting of 6 teachers and 3 paraeducators? Explain why or why not.

Answer 1

The probability of randomly selecting 3 paraeducators from a group of 6 teachers and 3 paraeducators is 1/84. Based on this probability, there is reason to doubt the Senior Class Sponsor’s claim that the selection was truly random. The proposed simulation process of rolling three dice does not correctly simulate the random selection of 3 paraeducators.

Step-by-step explanation:

To calculate the probability of randomly selecting 3 paraeducators from a group of 6 teachers and 3 paraeducators, we need to find the number of ways to select 3 paraeducators out of the 9 total volunteers, divided by the total number of ways to select any 3 volunteers.

The number of ways to select 3 paraeducators from a group of 3 paraeducators is 1. The number of ways to select 3 volunteers from a group of 9 total volunteers is 9 choose 3, which is equal to 84. So, the probability is 1/84, or approximately 0.0119.

(b) Based on this probability, there is reason to doubt the Senior Class Sponsor’s claim that the selection was truly random. The probability of randomly selecting 3 paraeducators out of the total 9 volunteers is quite low, which suggests that the selection process may have been biased or non-random.

(c) No, the proposed simulation process of rolling three dice does not correctly simulate the random selection of 3 paraeducators from the group of 9 volunteers. The simulation assumes that each die roll has an equal chance of selecting a teacher or a paraeducator. However, in reality, there are more teachers than paraeducators, so the probability of selecting a teacher is higher than the probability of selecting a paraeducator. Therefore, the simulation process does not accurately represent the true probabilities of selecting 3 paraeducators.