12 A computer science department has a probability of 0.35 that a senior receives a job offer in IT before graduation. Random select 8 senior students
• What is the probability that 5 students received offers before graduation?
• What is the probability only 1 student received offer before graduation?
• What is the probability that none of the 8 students received offers before graduation?
To find the probability of different outcomes, we will use the binomial probability formula. The formula for the probability of getting exactly k successes (in this case, receiving job offers before graduation) in n independent trials (in this case, randomly selecting senior students) is:
P(k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(k) is the probability of getting k successes
- C(n, k) is the number of combinations of n items taken k at a time (also known as the binomial coefficient)
- p is the probability of success in a single trial (in this case, receiving a job offer before graduation)
- (1-p) is the probability of failure in a single trial (in this case, not receiving a job offer before graduation)
- n is the total number of trials (in this case, the number of senior students randomly selected)
- k is the number of successes (in this case, the number of students who received job offers before graduation)
Now, let's calculate the probabilities for each question.
1. Probability that 5 students received offers before graduation:
Using the formula, we have:
P(5) = C(8, 5) * (0.35)^5 * (1-0.35)^(8-5)
Calculating:
P(5) = 56 * 0.35^5 * 0.65^3
P(5) ≈ 0.204
So, the probability that exactly 5 students received job offers before graduation is approximately 0.204, or 20.4%.
2. Probability that only 1 student received an offer before graduation:
Using the formula, we have:
P(1) = C(8, 1) * (0.35)^1 * (1-0.35)^(8-1)
Calculating:
P(1) = 8 * 0.35^1 * 0.65^7
P(1) ≈ 0.250
So, the probability that exactly 1 student received a job offer before graduation is approximately 0.250, or 25.0%.
3. Probability that none of the 8 students received offers before graduation:
Using the formula, we have:
P(0) = C(8, 0) * (0.35)^0 * (1-0.35)^(8-0)
Calculating:
P(0) = 1 * 0.35^0 * 0.65^8
P(0) ≈ 0.108
So, the probability that none of the 8 students received job offers before graduation is approximately 0.108, or 10.8%.