If a king is drawn from deck of 52 & then replaced what is probability another king will be drawn 4/52 X 4/52?

Question

would answer be 4/52 X 4/52 = 16/2704 or .0059

Also, if king is not replaced what is probability a king will be drawn 2nd selection would it be 4/52 X 3/51 = 12/ 2652 or.00452

also of 100 employees 57 are production workers, 40 are supervisors, 2 are secretaries and one is president If one employee is selected:

what is probability the selected emplyee is either a production worker or supervisor Answer: 57/100 + 40/100 = 97/100 or .97

are these events mutually exclusive?

what is the probability the slected employee is neither a production worker nor supervisor Answer 1- (57/100 + 40/100) 1- .97 = .03

Answer 1

I am not sure you have stated the question precisely.  

If the question is: What is the probability that on the first draw from a deck a king is drawn, then the king is put back into the deck, shuffled, then a second card is drawn and that too is a king. -  Then the answer is 4/52 x 4/52.  That is because each draw is an independent event and the total probability is the product of the two. The probability of each draw being a king is 4/52 for obvious reasons. This is exactly equal to the probability that you have two decks of cards and you draw one card from each and both are kings.

On the other hand if the question is: A card is drawn from a deck and found to be a king. The card is replaced and the deck is shuffled.  Then a card is drawn from the deck, what is the probability that this card is a king. - The answer to this is simply 4/52. This is because this question is only asking what the probability that one card drawn from a full random deck is a king.  The fact that a previous card is drawn is completely irrelevant.

Assuming that the question is the first one and altering the question with the king not being replaced the probability of drawing 2 kings in a row will  4/52 x 3/51 as you guessed. This is because the probability of a king on the first draw is 4/52 as you suggested and on the second is 3/51 for obvious reasons and since the two are independent events the total probability is the product of the two.

For your second event there are 2 ways of getting a successful outcome and they are clearly mutually exclusive. The two ways are 1) by picking a production worker and 2) by picking a supervisor.  Since you are only picking one worker you obviously cannot pick both a production worker and a supervisor.  This makes the two events mutually exlusive. Thus they are dependent events. Since they are dependent events the total probability is the sum of the probabilities of each event.

Probability of picking production worker = 57/100

Probability of picking supervisor = 40/100

Total probability of picking either a prod worker or supervisor = 57/100 + 40/100

The probability of picking neither a prod worker or supervisor is exactly the inverse of the former.  This is because every time you pick a worker it has to be one of:

1) it is a production worker or supervisor

2) it is neither a production worker or supervisor

In other words these 2 cases are collectively exhaustive - there is no possibility other than these 2.

Therefore the probability of 2) is the inverse of 1) and since we know what the probability of 1) is, the probability of 2) is:

1 - (57/100 + 40/100)

All your answers were correct (assuming I interpreted the question correctly).  I guess you were either just looking for a good explanation or confirming the answer.

Answer 2

Actually do this experiment to see.

Answer 3

I'm not sure I understand the first question.  If a king is drawn and then replaced, the probability that the next card drawn is a king is 4/52.  In other words it's the same probability as the first king.  

If you count the probability of the first king, then it's (4/52)*(4/52).  Meaning that if you drew a card at random, then put it back in the deck, then drew another card at random, that's the probability that BOTH cards will be kings.

If the king is not replaced, the probability of the second card being a king is 3/51 (3 kings left of 51 cards).

The probability of an employee being selected is either a production worker or supervisor is (57+40)/(57+40+2) or 97%.   The probability that the employee is neither a production worker or supervisor is 3/100 or 3%.  This is mutually exclusive with the other choice, and both add up to 100%.