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?
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To calculate the probabilities, we need to determine the total number of possible outcomes and the number of favorable outcomes.
a) To find the probability that three of the five students selected received an "A" for the final exam, we can use the concept of combinations.
The total number of possible outcomes is the number of ways to choose 5 students out of the 45 students in the class, which is represented as 45C5. This can be calculated as:
45C5 = 45! / (5! * (45-5)!) = (45 * 44 * 43 * 42 * 41) / (5 * 4 * 3 * 2 * 1) = 1,221,759
The number of favorable outcomes is the number of ways to choose 3 out of the 10 students who received an "A," which is represented as 10C3. This can be calculated as:
10C3 = 10! / (3! * (10-3)!) = (10 * 9 * 8) / (3 * 2 * 1) = 120
Therefore, the probability that three of the five randomly selected students received an "A" is:
P(3 A's) = favorable outcomes / total outcomes = 120 / 1,221,759 ≈ 0.0000983
b) To find the probability that at least one of the five students selected received an "A" for the final exam, we can calculate the probability of the complement event (the event of none of the selected students receiving an "A") and subtract it from 1.
The total number of possible outcomes remains the same as above, which is 1,221,759.
The number of favorable outcomes for none of the selected students receiving an "A" can be calculated as:
35C5 = 35! / (5! * (35-5)!) = (35 * 34 * 33 * 32 * 31) / (5 * 4 * 3 * 2 * 1) = 324,632
Therefore, the probability of none of the selected students receiving an "A" is:
P(no A's) = favorable outcomes / total outcomes = 324,632 / 1,221,759 ≈ 0.266
Finally, the probability of at least one of the five students selected received an "A" is:
P(at least 1 A) = 1 - P(no A's) = 1 - 0.266 = 0.734