What affect does a condition have on the probability? Let's compare and find out.
Example: Let's look at a standard deck of 52 playing cards.
No replacement: What is the probability of getting 2 kings assuming that the first one is not replaced? (Note: There are 4 kings in a deck of cards. And a deck of cards has 52 cards.)
Answer: 4/52 * 3/51 = 1/13*1/17= 1/221
Replacement: What is the probability of getting 2 kings assuming that the first one is replaced?
Answer: 4/52 * 4/52 = 1/13 * 1/13 = 1/169
Make up a similar example to the above problem using (a) with replacement and (b) without replacement. You may use playing cards, or a similar scenario, however be sure that you can represent both with and without replacement. Flipping a coin 10 times is not appropriate because it only represents the "without replacement" condition. Do not solve your own problem - that will be done by your classmates.
Sample Answer
A condition can significantly alter the probability of an event. When a condition is introduced, it often changes the total number of possible outcomes, the number of favorable outcomes, or both, thereby affecting the final probability. The key difference lies in whether the events are dependent or independent.
Dependent vs. Independent Events
Without replacement describes dependent events. The outcome of the first event affects the probability of the second event. This is because the total number of items available for the second draw changes. In your example of drawing two kings without replacement, the probability of drawing the second king is dependent on the first draw because there's one less king and one less card overall.
With replacement describes independent events. The outcome of the first event doesn't influence the probability of the second event. After the first item is selected, it's put back, so the total number of items and the number of favorable outcomes remain the same for the second draw. This is what you see in the example of drawing two kings with replacement; the probability of the second king is the same as the first.
Your Example Problem
A great example to illustrate both scenarios is drawing marbles from a bag.
Scenario: A bag contains 10 marbles: 5 red, 3 blue, and 2 green.
(a) With replacement: What is the probability of drawing two red marbles in a row?
(b) Without replacement: What is the probability of drawing a blue marble followed by a green marble?