Suppose you have
2 bags with the exact same sets of marbles inside.
Let's say there are 4 red, 5 blue and 9 yellow marbles in
From the first bag, you reach in and make a selection.
You record the color and then drop the marble back into the
bag. You then repeat the experiment a second time.
From the second bag you do exactly the same thing EXCEPT,
after you select the first marble and record it's color, you
do NOT put the marble back into the bag, You then
select a second marble, just like the other experiment.
The first experiment involves a process called
You put the object back into the bag so that the number of
marbles to choose from is the same for both draws.
The second experiment involves a process called
You do not put the object back in the bag so that the number
of marbles is one less than for the first draw.
As you might imagine, the probabilities for the 2
experiments will not be the same. In this lesson we
will illustrate a variety of these types of problems and
explain how to arrive at the correct solutions.
An Important Note
Sometimes a problem will not specifically state whether it is a
problem "with or without replacement". In these cases it
is very important to ask yourself this question:
"Is this problem with replacement?"
"Is this problem without replacement?"
Let common sense and a little intuition guide you through
types of problems.
A player is dealt 2 cards from a standard deck of 52 cards.
What is the probability of getting a pair of aces?
about it....obviously this is an experiment "without
replacement" because the player was given 2 cards.
To calculate the probability of a pair of aces you use the rules
for compound events:
P(ace on first card) = 4/52
(remember,there are 4 aces in the deck)
P(ace on the second card) = 3/51!
(the first card drawn was an ace!)
So the probability of getting 2 aces is:
P(ace,ace) = (4/52)(3/51) = 12/2652 =
A jar contains 2 red and 5 green marbles. A marble is
drawn, it's color noted, and put back in the jar. This
process is repeated a total of 4 times.
What is the probability that you selected 4 green marbles?
Since you put the marble back
in the jar after each selection this is an experiment "with
So,the probability for each draw will be exactly the same:
P(green) = 5/7
P(g,g,g,g) = 625/2401
Remember...Let common sense be your guide!!!