18.X. Binomial Distribution – Complete Detail

Try these first

1. 1.

   What is the probability of getting exactly 1 head when you flip a fair coin twice?

   Hint

   Use the PMF formula for n=2, k=1, and p=0.5.

2. 2.

   If a die is rolled 3 times, what is the probability of getting exactly 2 sixes?

   Hint

   Recognize there are 3 ways to choose which roll is not a six.

3. 3.

   What is the formula for the probability mass function of the Binomial Distribution?

   • P(X = k) = (n choose k) * p^(n-k) * (1-p)^k
   • P(X = k) = (n-k choose k) * p^k * (1-p)^n
   • P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

   Hint

   Remember the arrangement of p and (1-p) with respect to k and n.

4. 4.

   The mean of a binomial distribution is calculated as which of the following?

   • E(X) = n(1 - p)
   • E(X) = np
   • E(X) = p/n

   Hint

   Think about how the average number of successes is determined.

5. 5.

   A biased coin shows a 60% chance of landing heads. If flipped 10 times, what is the probability of getting exactly 6 heads?

   Hint

   Calculate the binomial coefficient and powers separately.

6. 6.

   In a shipment of 100 items, 90% are defect-free. If 15 items are chosen at random, compute the probability that at least 12 will be defect-free.

   Hint

   Consider using a cumulative approach for summation.