Question

From the first test, 65% of the class received an ‘A’ grade.

1. Sampling 5 students, what is the probability that 1 will have an A?

2. Sampling 6 students, what is the probability that all 6 will have an A?

3. Sampling 12 students, what is the probability that at LEAST 9 will have an A?

Answer 1

From the first test, 65% of the class received an ‘A’ grade.

This is a binomial experiment.

The following notation is helpful, when we talk about binomial probability.

• x: The number of successes that result from the binomial experiment.
• n: The number of trials in the binomial experiment.
• P: The probability of success on an individual trial.
• Q: The probability of failure on an individual trial. (This is equal to 1 - P.)
• n!: The factorial of n (also known as n factorial).
• b(x; n, P): Binomial probability - the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P.
• : The number of combinations of n things, taken r at a time

Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:

b(x; n, P) = * P^x * (1 - P)^{n - x}

Here, Let getting A grade be considered as success P is the probability of success on an individual trial.

Given, P = 65/100 = 0.65 ; Q =1-0.65 =  0.35

1. Sampling 5 students, what is the probability that 1 will have an A?

Here, n = 5 , x = 1

b(1, 5, 0.65) =   = 0.04877

2. Sampling 6 students, what is the probability that all 6 will have an A?

Here, n = 6 , x = 6

b(6, 6, 0.65) = = 0.07541

3. Sampling 12 students, what is the probability that at LEAST 9 will have an A?

Here,n = 12, x9

b(x9, 12, 0.65) =

=0.1953+0.1088+0.0367+0.00568 = 0.34648

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