Question 845125: Mrs. Moore's class is conducting an experiment to find the probability of pulling certain colors from a bag of 25 marbles. If 2 are purple, 2 are yellow, 7 are green, and the rest are black, what is the probability of drawing 2 black if the marbles are not replaced after they are picked? Show answer as a percent rounded to nearest hundredth, if necessary.
Answer by pmesler(52)   (Show Source):
You can put this solution on YOUR website! When it comes to probability, you first have to consider something called the probability space or P(S). The probability space is simply the total number of outcomes. In probability we say that the probability that an event E will occur is the number of favorable outcomes / the number of total outcomes.
We write the number of total outcomes as P(S) which is called the probability space. In other words P(E)/P(S) is the probability that P(E), or some favorable outcome will occur.
Now going back to the problem. We know that there are 2 purple, 2 yellow and 7 green marbles. The rest are black. Let's find out first how many black marbles there are. If we add up the numbers for the purple, yellow, and green marbles we get 11 marbles. The remaining black marbles must then be 25-11 = 14.
We can rewrite the marbles and their numbers like this
Purple P(p) = 2
Yellow P(y) = 2
Green P(g) = 7
Black P(b) = 14.
Now we need to calculate the probability of drawing two black marbles from a bag of 25 marbles. But before we do that, it's important to make note of something the problem tells us. It says that "the marbles are not replaced." What that means is that the outcome of the second event is affected by the outcome of the first event. We say that these two events are DEPENDENT.
We'll say that the probability of pulling a black marble the first time is P(b1) and the probability of pulling a second marble is P(b2). Since there are 14 black marbles and the total number of marbles is 25 we write
P(b1)/P(S) = 14/25 = .56 or 56%.
Therefore you have a 56% chance of drawing a black marble on the first try. Now, because a black marble is not replaced, we have reduced the total number of black marbles and the total number of all marbles at the same time. Therefore the probability that you will pull a black marble from the bag a second time is
P(b2)/P(S) = 13/24.
Now, because the two events are dependent on each other, dependent because pulling out a black marble the first time reduces the probability of extracting a black marble the second time, we multiply the two probabilities to find the probability of these two DEPENDENT events.
14/25 * 13/24 = 182/600 = .3033 or 30.33%
Therefore the probability of drawing 2 black marbles if none are replaced after they are picked is 30.33%.
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