Step-by-step explanation:
A sack of marbles contains 9 red, 7 green, and 5 blue marbles. If we pull five marbles without replacement, what is the probability that exactly one is green?
The probability of an event occurring is the ratio of the number of ways it can occur to the total number of events in the sample space.
Let’s pretend for a moment that each of the marbles is labeled with a number in addition to its color, so each is distinct. Note that this doesn’t change the probability of the event in question.
First, how many ways are there for the set of marbles drawn to include exactly one green marble? Well, first we would need to choose 1 of the green marbles from the set of 7 , and then in each of those cases, we would need to choose 4 other marbles from the remaining 9+5=14 . Using binomial coefficients to represent these combinations[1] , this gives us
(71)(144)
ways to select exactly one green marble.
Now, how many total possibilities are there? This is just the number of combinations of 5 marbles that can be drawn from a set of 9+7+5=21 . Thus, there are
(215)
total possibilities.
With that, we have the probability as
Ways to draw exactly one green marble in a set of fiveWays to draw any five marbles
=(71)(144)(215)
=10012907≈34.43%
