A bag has five blue marbles,
four green marbles, and seven red marbles,
as pictured here below.
Notice how we have a total of five plus four plus seven
or 16 marbles in the bag.
For number one,
you select two marbles one at a time with replacement,
which means you select one marble, look at the color,
replace the marble back into the bag,
and then select the second marble.
Determine the probability the first marble is blue
and the second marble is also blue.
So if we let B equal the selection of a blue marble,
we can say the probability is equal to the probability of B
and then B again.
And because the first marble is selected
and then replaced back into the bag
before the second marble is selected,
the outcome of one event
does not affect the outcome of the other.
And therefore, the two events are independent.
Independent events are events where an outcome of one event
is not affected by the outcome of the other event.
If events A and B are independent,
then the probability of both A and B occurring
is equal to the probability of A times the probability of B.
So in our case,
the probability of B and B is equal to the probability of B
times the probability of B
because the events are independent,
because we are selecting marbles with replacement.
And now let's determine the probability of B.
Because five of the 16 marbles are blue,
the probability of selecting a blue marble
is five out of 16, or 5/16,
which gives us 5/16 times,
again, the probability of selecting a blue marble is 5/16.
So we have 5/16 times 5/16.
It then simplifies.
And if we're multiplying, we have 25, 256,
or 25 divided by 256.
Let's also express this as a decimal and a percentage.
To four decimal places,
the fraction is approximately 0.0977,
which is 9.77%.
And now for the second example,
you select two marbles one at a time without replacement,
which means you select the first marble,
do not replace it back into the bag,
and then select the second marble.
We were asked to determine the probability
the first marble is blue
and again the second marble is also blue.
So we want the probability of a blue marble
and another blue marble, but now without replacement.
So because the first marble
is not replaced back into the bag,
the outcome of one event is affected
by the outcome of the other.
And therefore, now we have dependent events,
where dependent events are events where an outcome
of one event is affected by the outcome of the other event.
And the probability that event B occurs
given event A has happened
is represented using this notation here,
which is read the probability of B given A has occurred.
So if events A and B are dependent,
the probability of both A and B occurring
is equal to the probability of A times
the probability of B given A has occurred.
So again, for our example, because we have dependent events,
the probability of B and B is equal to the probability of B
times the probability of B
given B has occurred.
So we know from part A,
the probability of selecting a blue marble is 5/16.
But in this case, a blue marble is selected
and not replaced back into the bag
before we select the second marble.
So let's just say this is the first blue marble selected.
Because it's not replaced back into the bag,
now we have a total of 15 marbles.
And now only four of the 15 marbles are blue.
And therefore, the probability of selecting a blue marble
given a blue marble has already been selected
and not replaced is four out of 15, or 4/15.
So notice how the number of favorable outcomes
has decreased by one,
and the total number of marbles has also decreased by one.
Before multiplying, notice how we can simplify.
There's a common factor of four between four and 16.
There's one four in four and four fours in 16.
There's also a common factor of five between five and 15.
There's one five in five and three fives in 15.
Multiplying, the probability is 1/12.
So the probability of selecting a blue marble
and then another blue marble without replacement is 1/12.
Let's also show the decimal and percentage.
To four decimal places, 1/12 is approximately 0.0833,
which 8.33%.
I hope you found this helpful.