T and M are events such that n(U)=30, n(M)=17, n(T)=10, n((TUM)')=5
Draw a Venn diagram to display this information
Hence find P(T and M)
find P((T and not M)given that M)
My Venn consists of two intersecting circles within a rectangle, rectangle is 30
from your data, I let the intersection of M and T be x
Then the part of only M is 17-x, and the part of only T is 10-x
n((TUM)')=5 --> The part outside the two circles but within the rectangle is 5
17-x + x + 10-x + 5 =30
x = 2
Now you know everything.
Oh boy, let's get started on this Venn diagram party! Here's a visual representation of the information:
_______
| |
T | M |
|_______|
Now let's delve into the probability shenanigans:
To find P(T and M), we need to find the probability of both events T and M happening together. Looking at the diagram, we see that the overlapping region represents the intersection of T and M. Unfortunately, the sizes of these regions aren't given, so we can't calculate the exact probability here.
Now onto P((T and not M) given that M). This asks for the probability of T happening and M not happening, given that M has occurred. Since M has occurred, we can focus on the region representing M. We know that the size of (TUM)' is 5. Therefore, our focus should be on the remaining part of T within the region of M, which is the complement of (TUM)'.
Again, the exact sizes of the regions are missing, so we can't give you an exact probability. But hey, life is full of surprises, right?
To draw the Venn diagram, we will need the following information:
n(U) = 30 (Total number of elements in the universal set)
n(M) = 17 (Number of elements in set M)
n(T) = 10 (Number of elements in set T)
n((TUM)') = 5 (Number of elements outside the set (TUM))
First, draw a rectangle to represent the universal set U. Label it U and write the value 30 inside.
Next, draw two overlapping circles within the rectangle to represent sets T and M. Label them T and M respectively. Write the values 10 and 17 inside the circles to represent n(T) and n(M).
Finally, outside the intersection of sets T and M, write the value 5 to represent n((TUM)').
The diagram should now show three distinct regions: T only, M only, and the intersection of T and M. The values inside the circles and outside the intersection point should match the given information.
To find P(T and M) (the probability of both T and M occurring), we divide the number of elements in the intersection of T and M by the total number of elements in the universal set:
P(T and M) = n(T ∩ M) / n(U)
Since we don't have the value of n(T ∩ M) directly, we can use the information given to calculate it. We know that n((TUM)') = 5, which represents the number of elements outside the set (TUM). Therefore:
n((TUM)') = n(U) - n(T ∪ M)
Rearranging the equation, we get:
n(T ∩ M) = n(U) - n((TUM)')
Substituting the given values, we have:
n(T ∩ M) = 30 - 5 = 25
Now we can calculate P(T and M):
P(T and M) = n(T ∩ M) / n(U) = 25 / 30 = 5/6 or approximately 0.8333
To find P((T and not M) given that M) (the probability of T occurring but not M, given that M has already occurred), we divide the number of elements in the set (T and not M) intersected with M by the total number of elements in M:
P((T and not M) given that M) = n((T ∩ M') ∩ M) / n(M)
Since we don't have the value of n((T ∩ M') ∩ M) directly, we can use the equation:
n((T ∩ M') ∩ M) = n(M) - n(T ∩ M)
Substituting the given values, we have:
n((T ∩ M') ∩ M) = 17 - 25 = -8 (negative value indicates an empty set)
Since we have an empty set, the probability P((T and not M) given that M) is 0.
Therefore:
P((T and not M) given that M) = 0
To draw the Venn diagram, start by drawing a rectangle to represent the universal set U, which has a total of 30 elements. Label this rectangle as U.
Inside the rectangle, draw two overlapping circles to represent events T and M. Label the circle for T as T and the circle for M as M.
Label the intersection of T and M as TUM. Since n((TUM)') = 5, this means that there are 5 elements outside the intersection of T and M. Place the number 5 outside the overlapping section of T and M.
Label the remaining sections outside TUM as T', M', and U' accordingly. We do not know the exact number of elements in each of these sections from the given information, so leave them unlabeled for now.
To find P(T and M), divide the number of elements in the intersection of T and M (n(TUM)) by the total number of elements in the universal set (n(U)). In this case, P(T and M) = n(TUM) / n(U) = 10 / 30 = 1/3.
To find P((T and not M) given that M), we need to find the probability of T occurring while M does not occur, given that M has occurred. Mathematically, this can be represented as P(T and not M | M). The information provided does not give us direct numbers to calculate this probability, so further data or assumptions are needed to determine this probability.