Multiplication Theorem on Probability

A box contains 10 red balls and 15 green balls. Two balls are drawn in succession without replacement. What is the probability that, the first is red and the second is green?

A bag contains 8 red balls and 5 white balls. Two successive draws of 3 balls are made without replacement. Find the probability that the first drawing will give 3 white balls and second drawing will give 3 red balls.

A company produces 10,000 items per day. On a particular day, 2500 items were produced on machine A, 3500 on machine B, and 4000 on machine C. The probability that an item is produced by the machines. A, B, C to be defective is respectively 2%, 3%, and 5%. If one item is selected at random from the output and is found to be defective, then the probability that it was produced by machine C, is ______ 

A card is drawn from a well-shuffled pack of 52 cards. Consider two events A and B as
A: a club 6 card is drawn.
B: an ace card 18 drawn.
Determine whether the events A and B are independent or not.

A bag contains 5 white and 4 black balls and another bag contains 7 white and 9 black balls. A ball is drawn from the first bag and two balls are drawn from the second bag. What is the probability of drawing one white and two black balls? 

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

Let A and B be two events such that P(A) = 0.6, P(B) = 0.2, and P(A|B) = 0.5. Then P(A′|B′) equals ______.

Read the following passage and answer the questions given below:

In an Office three employees Jayant, Sonia and Oliver process incoming copies of a certain form. Jayant processes 50% of the forms, Sonia processes 20% and Oliver the remaining 30% of the forms. Jayant has an error rate of 0.06, Sonia has an error rate of 0.04 and Oliver has an error rate of 0.03.

Based on the above information, answer the following questions.

1. Find the probability that Sonia processed the form and committed an error.
2. Find the total probability of committing an error in processing the form.
3. The manager of the Company wants to do a quality check. During inspection, he selects a form at random from the days output of processed form. If the form selected at random has an error, find the probability that the form is not processed by Jayant.
   OR
   Let E be the event of committing an error in processing the form and let E₁, E₂ and E₃ be the events that Jayant, Sonia and Oliver processed the form. Find the value of `sum_(i = 1)^3P(E_i|E)`.