1) A group of students were asked about their itunes collection. The probability that a student had downloaded the album by group X was 33%. The probability that a student downloaded the album by group Y was 41%. However, if you only asked the students that downloaded the X album, there was a probability of 59% that they would also have the group Y album (think conditional probability). Draw a Venn Diagram ( X and Y will intersect)
Find:
a)The probability that a student, chosen at random, will have both albums dowloaded. Think ∩ .
b) A student with the group Y album, will also have the group X album
c) A student has the group X album, but not the group Y album.
d) A student has neither album.(think intersection of compliments
To solve this problem and understand the probability of various scenarios, let's start by creating a Venn diagram based on the given information:
1) Draw two overlapping circles to represent group X and group Y.
2) Label the intersection of the circles as the area where both group X and group Y albums exist. This represents students who have downloaded both albums.
3) Assign the probability of owning the group X album to 33% and the probability of owning the group Y album to 41%.
4) Within the intersection area, assign a probability of 59% to represent the percentage of students who downloaded the group X album and also have the group Y album.
5) Outside the intersection area, label the remaining portions of the circles as follows:
- The portion only in the X circle represents students who have the group X album but not the group Y album.
- The portion only in the Y circle represents students who have the group Y album but not the group X album.
- The portion outside both circles represents students who have neither album.
Now, let's answer the questions:
a) The probability that a student chosen at random will have both albums downloaded is the probability of the intersection of both events. In the Venn diagram, this corresponds to the shared area of the two circles. Therefore, the probability is 59% based on the given information.
b) To find the probability that a student with the group Y album also has the group X album, we only need to consider the Y circle. Since 59% of those who have the group X album also have the group Y album, we can conclude that the probability is 59%.
c) To find the probability that a student has the group X album but not the group Y album, we need to look at the portion only in the X circle. Since the total probability of having the group X album is 33%, and the portion overlapping with the Y circle is 59%, we subtract this overlap from the total probability: 33% - 59% = -26%. However, negative probabilities have no practical meaning, so we would consider the probability as 0%.
d) To find the probability that a student has neither album, we subtract the total probability of having both albums, having only the X album, and having only the Y album from 100% (since the sum of probabilities must equal 1 or 100%). In this case, the probability of having both albums is 59%, the probability of having only the X album is 33%, and the probability of having only the Y album is 41%, so the probability of having neither album is 100% - 59% - 33% - 41% = -33%. Similar to the previous scenario, negative probabilities have no practical meaning, so we would consider the probability as 0%.
Therefore, the answers to the questions are:
a) The probability that a student, chosen at random, will have both albums downloaded is 59%.
b) A student with the group Y album will also have the group X album with a probability of 59%.
c) The probability that a student has the group X album but not the group Y album is 0%.
d) The probability that a student has neither album is 0%.