A class consists of 45 students. Ten of these students received an "A" for the final exam. Five
students are selected at random from this group.
a) What is the probability that three of the five students selected received an "A" for the final
exam?
b) What is the probability that at least one of the five students selected received an "A" for the
final exam?
To find the probability in these scenarios, we need to use the concept of combinations and calculate the number of favorable outcomes over the total number of possible outcomes.
a) What is the probability that three of the five students selected received an "A" for the final exam?
To calculate this probability, we need to find the number of ways to choose three students from the ten who got an "A" and two students from the remaining 35 who did not get an "A." We will then divide this by the total number of ways to choose five students from the entire class.
The number of favorable outcomes is calculated as follows:
Number of ways to choose three students from the ten who received an "A": C(10, 3) = 10! / (3! * (10-3)!) = 120.
Number of ways to choose two students from the remaining 35 who did not receive an "A": C(35, 2) = 35! / (2! * (35-2)!) = 595.
The total number of possible outcomes is the number of ways to choose five students from the entire class: C(45, 5) = 45! / (5! * (45-5)!) = 122,175.
So, the probability that three of the five students selected received an "A" for the final exam is:
P(3 students with an "A") = (Number of favorable outcomes) / (Total number of possible outcomes)
= (120 * 595) / 122,175
= 7,140 / 122,175
≈ 0.0584 (rounded to four decimal places)
b) What is the probability that at least one of the five students selected received an "A" for the final exam?
To calculate this probability, we can find the probability of the complementary event, which is when none of the five students received an "A." Then, we subtract this probability from 1 to get the probability of at least one student receiving an "A."
The number of favorable outcomes for the complementary event is the number of ways to choose five students from the 35 who did not receive an "A": C(35, 5) = 35! / (5! * (35-5)!) ≈ 324,632.
So, the probability that at least one of the five students selected received an "A" for the final exam is:
P(at least one student with an "A")) = 1 - P(none of the five students received an "A")
= 1 - (Number of favorable outcomes for complementary event) / (Total number of possible outcomes)
= 1 - 324,632 / 122,175
≈ 1 - 2.66
≈ -1.66
Note: The probability cannot be negative or greater than 1. Please recheck the numbers provided or let me know if you made any error.