A bag contains six blue balls and 15 red balls. If two balls are drawn without replacement, what is the probability of getting one blue ball and one red ball?
Our bag has six blue and 15 red for a total of 21 balls. To find the probability of getting one red and one blue, we’ll have to consider two cases. The first case is drawing red and then blue, and the second case is drawing blue, then red.
Let’s start by considering the probability of drawing red and then blue. We need to remember here that we are drawing these without replacement. The probability of drawing red on the first draw is 15 out of 21. But if we take one ball away and we do not replace it, then, on our second draw, we’ll only have 20 balls to choose from. And if we’re going to choose a blue ball from that 20, there are six remaining blue balls. To find the probability of these two events, you then multiply them together.
To make this multiplication simpler, we can simplify before we multiply. 15 over 20 can be simplified to three-fourths. And three divided by three is one. 21 divided by three is seven. One final thing we can reduce, we can divide six by two to get three and four by two to get two. Three times one is three. Seven times two is 14. The probability of choosing a red and then a blue is three fourteenths.
We want to follow the same procedure. This time we’ll consider the blue first, then the red. If we start back at the beginning, there are 21 total balls and six of them are blue. That’s the probability of getting a blue ball on the first turn, six out of 21. Again, we do not replace it. And so there are only 20 to choose from in the second draw. And out of this 20, 15 of them are red.
To find out what these probabilities are together, we would multiply them. But if we look closely, we’ll see that these two probabilities are the same because when we multiply fractions, we multiply their numerators. And 15 times six is the same as six times 15. And then we multiply the denominators. 21 times 20 is the same thing as 21 times 20, which makes the probability of selecting blue and then red also three fourteenths.
But when we’re talking about probability and we’re talking about either/or, that is the union of these two probabilities. And to find the union of these two events, the probability that we would get one red and one blue, we would need to add these two probabilities together. Three fourteenths plus three fourteenths, which is six fourteenths or in a simplified form three-sevenths.