Approimately 10.3% of American high school students drop out of school before gradution. Choose 10 students entering high school at random. Find the probability that (Assume binomial distribution): a. Exactly two drop out b. At least 7 graduate c. All ten stay in school and graduate
Answer to the question
To calculate the probabilities using a binomial distribution, we will use the following 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,
n is the number of trials or students (in this case, 10),
k is the number of desired successes or dropouts/graduates,
p is the probability of success (dropout rate in this case), and
C(n, k) denotes the number of combinations of n items taken k at a time.
a. Exactly two drop out:
Using the formula, plug in the values:
P(X=2) = C(10, 2) * (0.103)^2 * (1-0.103)^(10-2)
Calculations:
P(X=2) = (10!/(2!(10-2)!)) * (0.103)^2 * (0.897)^8
P(X=2) = 45 * (0.010609) * (0.43046721)
P(X=2) ≈ 0.199
Therefore, the probability that exactly two out of ten students drop out is approximately 0.199.
b. At least seven graduate:
To calculate this probability, we need to find the probabilities of having 7, 8, 9, or 10 students graduate and then sum them up.
P(X≥7) = P(X=7) + P(X=8) + P(X=9) + P(X=10)
Calculations:
P(X=7) = C(10, 7) * (0.103)^7 * (0.897)^3
P(X=8) = C(10, 8) * (0.103)^8 * (0.897)^2
P(X=9) = C(10, 9) * (0.103)^9 * (0.897)^1
P(X=10) = C(10, 10) * (0.103)^10 * (0.897)^0
P(X≥7) = P(X=7) + P(X=8) + P(X=9) + P(X=10)
Calculations:
P(X=7) = (10!/(7!(10-7)!)) * (0.103)^7 * (0.897)^3
P(X=8) = (10!/(8!(10-8)!)) * (0.103)^8 * (0.897)^2
P(X=9) = (10!/(9!(10-9)!)) * (0.103)^9 * (0.897)^1
P(X=10) = (10!/(10!(10-10)!)) * (0.103)^10 * (0.897)^0
P(X≥7) = P(X=7) + P(X=8) + P(X=9) + P(X=10)
Calculations:
P(X≥7) ≈ 0.139 + 0.035 + 0.005 + 0.0001
P(X≥7) ≈ 0.1791
Therefore, the probability of at least seven out of ten students graduating is approximately 0.1791.
c. All ten stay in school and graduate:
The probability that all ten students stay in school and graduate can be calculated using the formula:
P(X=0) = C(10, 0) * (0.103)^0 * (1-0.103)^(10-0)
Calculations:
P(X=0) = (10!/(0!(10-0)!)) * (0.103)^0 * (0.897)^10
P(X=0) ≈ (1) * (1) * (0.23347444)
P(X=0) ≈ 0.233
Therefore, the probability that all ten students stay in school and graduate is approximately 0.233.