There are 49 mice in a pet shop.

30 mice are white.
27 mice are male.
18 mice have short tails.
8 mice are white and have short tails.
11 mice are male and have short tails.
7 mice are male but neither white nor short-tailed.
5 mice have all three characteristics and
2 have none.

Note:
W represents white mice
M represents male mice
A represents short-failed mice

I made a venn diagram but need help on the following:

1. n(M upside down u W)
2. n(M' U S)
3. P( both mice are short-tailed)

My teacher did not go over these notions so it would be helpful if you explain what it means and how you derive the numbers.....

1. 14

2. 7
3. 51/392

1. n(M ∪ W) represents the number of mice that are either male or white or both. To find this value, we need to add the number of male mice (27) with the number of white mice (30) and then subtract the number of mice that are both male and white (8) because we don't want to count them twice.

n(M ∪ W) = n(M) + n(W) - n(M ∩ W)
= 27 + 30 - 8
= 49

So, there are 49 mice that are either male or white or both.

2. n(M' ∪ A) represents the number of mice that are either not male or have short tails or both. To find this value, we need to subtract the number of male mice (27) from the total number of mice (49) and then add the number of mice that have short tails (18) because we want to include those that have short tails but are not male.

n(M' ∪ A) = n(Total) - n(M) + n(A)
= 49 - 27 + 18
= 40

So, there are 40 mice that are either not male or have short tails or both.

3. P(both mice are short-tailed) represents the probability that two randomly chosen mice have short tails. To find this probability, we need to find the number of mice that have short tails (18) and then divide it by the total number of mice (49) multiplied by the number of mice minus one (48), which represents the possible combinations of choosing two mice from the total population.

P(both mice are short-tailed) = n(A) / (n(Total) * (n(Total) - 1))
= 18 / (49 * 48)
= 18 / 2352
≈ 0.0077

So, the probability that both mice are short-tailed is approximately 0.0077.

To answer the questions, let's start by interpreting the given information and creating a Venn diagram:

W represents white mice (30)
M represents male mice (27)
A represents mice with short tails (18)
8 mice are white and have short tails (W ∩ A = 8)
11 mice are male and have short tails (M ∩ A = 11)
7 mice are male but neither white nor short-tailed (M - (M ∩ W) - (M ∩ A) = 7)
2 mice have none of the characteristics (Not (W ∪ M ∪ A) = 2)

1. n(M upside down u W)
n(M upside down u W) represents the number of mice that are male or white. In the Venn diagram, n(M upside down u W) is the sum of the number of mice in the regions M, W, and the overlapping region between M and W. Calculating the value:
n(M upside down u W) = M + W - (M ∩ W)
n(M upside down u W) = 27 + 30 - 0 (since the overlapping region is not given)

Therefore, n(M upside down u W) = 57.

2. n(M' U S)
n(M' U S) represents the number of mice that are not male or have short tails. In the Venn diagram, n(M' U S) is the sum of the number of mice in the regions outside of M and the region representing short-tailed mice. Calculating the value:
n(M' U S) = Total number of mice - (M + A - (M ∩ A))
n(M' U S) = 49 - (27 + 18 - 11) (since the overlapping region between M and A is given)
n(M' U S) = 49 - 16

Therefore, n(M' U S) = 33.

3. P(both mice are short-tailed)
P(both mice are short-tailed) represents the probability that two randomly selected mice from the pet shop would both have short tails. To find the probability, we need to calculate the number of mice with short tails and divide it by the total number of mice:
P(both mice are short-tailed) = (Number of mice with short tails / Total number of mice) * (Number of mice with short tails - 1 / Total number of mice - 1)

P(both mice are short-tailed) = (18 / 49) * (17 / 48)

Therefore, P(both mice are short-tailed) ≈ 0.1053 or 10.53%.