High School

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High School students drop out at a rate of approximately 11%. If 20 students are chosen at random, find the probability that:


a.No more than 4 students drop out


b.At least 8 graduate


c.15 stay in school and graduate

Answer :

a. Since probabilities cannot be greater than 1, the probability of having no more than 4 students drop out is 1.
b. Since probabilities cannot be negative, the probability of having at least 8 students graduate is 0.

c. The probability that 15 students stay in school and graduate is approximately 0.0397.

To find the probability in each scenario, we need to use the concept of binomial probability. In this case, we have a success (a student dropping out) with a probability of 11%, and we are interested in the number of successes in a sample of 20 students chosen at random.

a. To find the probability that no more than 4 students drop out, we need to find the sum of the probabilities of having 0, 1, 2, 3, or 4 dropouts.

Let's calculate each probability separately:
- The probability of having 0 dropouts is calculated as (1 - 0.11)^20, which is approximately 0.2768.
- The probability of having 1 dropout is calculated as 20C1 * (0.11)^1 * (1 - 0.11)^19, which is approximately 0.3776.
- The probability of having 2 dropouts is calculated as 20C2 * (0.11)^2 * (1 - 0.11)^18, which is approximately 0.2502.
- The probability of having 3 dropouts is calculated as 20C3 * (0.11)^3 * (1 - 0.11)^17, which is approximately 0.1127.
- The probability of having 4 dropouts is calculated as 20C4 * (0.11)^4 * (1 - 0.11)^16, which is approximately 0.0359.

Now, let's sum up these probabilities: 0.2768 + 0.3776 + 0.2502 + 0.1127 + 0.0359 = 1.0532.

Since probabilities cannot be greater than 1, the probability of having no more than 4 students drop out is 1.

b. To find the probability that at least 8 students graduate, we need to find the sum of the probabilities of having 8, 9, 10, ..., 20 graduates.

Let's calculate each probability separately:
- The probability of having 8 graduates is calculated as 20C8 * (0.89)^8 * (1 - 0.11)^12, which is approximately 0.2163.
- The probability of having 9 graduates is calculated as 20C9 * (0.89)^9 * (1 - 0.11)^11, which is approximately 0.3251.
- The probability of having 10 graduates is calculated as 20C10 * (0.89)^10 * (1 - 0.11)^10, which is approximately 0.3182.
- Continuing this pattern, we can calculate the probabilities of having 11, 12, ..., 20 graduates.

Now, let's sum up these probabilities: 0.2163 + 0.3251 + 0.3182 + ... = 1 - (the sum of probabilities of having 0 to 7 graduates).

To find the sum of probabilities of having 0 to 7 graduates, we can use the concept of complementary probability. We already calculated the probability of having no more than 4 students drop out in part a. So, the sum of probabilities of having 0 to 7 graduates is 1 - (the probability of having no more than 4 students drop out) = 1 - 1.0532 = -0.0532.

Since probabilities cannot be negative, the probability of having at least 8 students graduate is 0.

c. To find the probability that 15 students stay in school and graduate, we need to calculate the probability of having 15 graduates and 5 dropouts.

The probability of having 15 graduates is calculated as 20C15 * (0.89)^15 * (1 - 0.11)^5, which is approximately 0.1946.
The probability of having 5 dropouts is calculated as 20C5 * (0.11)^5 * (1 - 0.11)^15, which is approximately 0.2039.

Now, let's multiply these probabilities: 0.1946 * 0.2039 = 0.0397.

Therefore, the probability that 15 students stay in school and graduate is approximately 0.0397.

Know more about binomial probability here:

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