Calculating the Probability of Exactly 2 Students Passing an Examination
The probability of a student passing a certain examination is 3/4. We are interested in finding the probability that out of 5 students selected at random, exactly 2 will pass the examination. This problem can be solved using the principles of binomial distribution and the binomial probability formula.
Understanding Binomial Distribution
Binomial distribution is a statistical method used to model the number of successes in a specific number of independent trials. Each trial has only two possible outcomes: success or failure. In this case, the success is the student passing the exam, and the failure is the student failing the exam. The binomial distribution is characterized by two parameters: the number of trials (n) and the probability of success in each trial (p).
Applying the Binomial Probability Formula
Given the problem details, we can use the binomial probability formula:
Formula: ( P(X k) binom{n}{k} p^k (1 - p)^{n - k} )
Interpreting the Parameters
n 5: Total number of students (trials). k 2: Number of students expected to pass (successes). p 3/4: Probability of a student passing the examination. 1 - p 1/4: Probability of a student failing the examination.Calculating the Probability
To calculate the probability that exactly 2 out of 5 students will pass the examination, we follow these steps:
Calculate the binomial coefficient: (binom{5}{2}) Calculate (p^k): (left(frac{3}{4}right)^2) Calculate (1 - p^{n - k}): (left(frac{1}{4}right)^3)Let's go through each step in detail:
Step 1: Calculate the Binomial Coefficient
(binom{5}{2} frac{5!}{2!(5-2)!} frac{5 times 4}{2 times 1} 10)
Step 2: Calculate (p^k)
(left(frac{3}{4}right)^2 frac{9}{16})
Step 3: Calculate (1 - p^{n - k})
(1 - left(frac{1}{4}right)^3 1 - frac{1}{64} frac{63}{64})
Now, substitute these values into the binomial probability formula:
Probability:
(P(X 2) binom{5}{2} left(frac{3}{4}right)^2 left(frac{1}{4}right)^3)
( 10 times frac{9}{16} times frac{1}{64})
( 10 times frac{9}{1024} frac{90}{1024} frac{45}{512})
Therefore, the probability that exactly 2 out of 5 students will pass the examination is:
Probability (frac{45}{512})
Expressed as a decimal, this is approximately 0.0878.
Conclusion
The problem described is an excellent example of a binomial distribution, where we calculated the probability of exactly 2 successes (students passing) out of 5 trials (students). The key to solving such problems lies in correctly identifying the parameters and applying the binomial probability formula.