Graduation rate for students in a program is 80%. If 10 students from it are selected at random, find the probability that AT LEAST 8 of them graduated.
To find the probability of at least 8 out of 10 students graduating, we need to calculate the probability of exactly 8, exactly 9, and exactly 10 students graduating, and then sum them up.
Let's begin by calculating the probability of exactly 8 students graduating. We can use the binomial probability formula:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting exactly k successes
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success (graduating rate in this case)
- n is the number of trials or students selected
In this case, n = 10 (10 students selected), p = 0.80 (graduation rate), and k = 8 (exactly 8 students graduating).
Using the formula, we have:
P(X=8) = C(10, 8) * 0.80^8 * (1-0.80)^(10-8)
Calculating this, we have:
P(X=8) = 45 * 0.80^8 * 0.20^2
Now, repeat the same process to find the probability of exactly 9 students graduating (k = 9) and exactly 10 students graduating (k = 10). Finally, sum up these probabilities to find the probability of at least 8 students graduating:
P(X >= 8) = P(X=8) + P(X=9) + P(X=10)
So, the probability that AT LEAST 8 out of 10 students graduated can be found by calculating the sum of these individual probabilities.