At a school fair, there are 4 bags to pick from. One bag has a prize, and the other three bags have nothing. If 6 students each randomly pick from one of the 4 bags, what is the probability that at least 3 students will get a prize?
To find the probability that at least 3 students will get a prize, we need to calculate the probability that exactly 3 students, exactly 4 students, exactly 5 students, or all 6 students pick the bag with the prize.
Let's calculate each of these probabilities:
1. Probability that exactly 3 students get a prize:
The number of ways to choose 3 students out of 6 is given by the combination formula: C(6,3) = 20.
For each combination of 3 students, there is a 1/4 probability that all 3 pick the bag with the prize, and the other 3 pick empty bags. Therefore, the probability for each combination is (1/4)^3 * (3/4)^3.
So, the total probability that exactly 3 students get a prize is 20 * (1/4)^3 * (3/4)^3 ≈ 0.08789.
2. Probability that exactly 4 students get a prize:
The probability that exactly 4 students get a prize is zero because there is only one prize and only 3 students will get a prize at most.
3. Probability that exactly 5 students get a prize:
The probability that exactly 5 students get a prize is also zero because there is only one prize and only 3 students will get a prize at most.
4. Probability that all 6 students get a prize:
The probability that all 6 students get a prize is (1/4)^6 ≈ 0.00024.
Now add up the probabilities for each scenario to find the total probability that at least 3 students will get a prize:
0.08789 + 0 + 0 + 0.00024 ≈ 0.08813
Therefore, the probability that at least 3 students will get a prize is approximately 0.08813 or 8.813%.