To solve this problem, we can use set notation and formulas. Let's break it down step by step.
First, let's use the given information to fill in the values in the Venn diagram:
- 10 paintings are neither T (from the 20th century) nor B (British).
- x paintings are both T and B.
- x(x-2) paintings are T (this represents the overlap between T and not B).
- 2x+8 paintings are B.
Now, we can set up equations to represent the information:
Total number of paintings = 60
10 (neither T nor B) + x(x-2) (T but not B) + x (both T and B) + 2x+8 (B) = 60
Simplifying the equation, we get:
x^2 - 2x + x + 2x + 8 + 10 = 60
x^2 + 11x + 18 = 60
x^2 + 11x - 42 = 0
Now, we can solve for x by factoring or using the quadratic formula. In this case, we can factor the equation as follows:
(x + 14)(x - 3) = 0
So, the possible values for x are -14 and 3. However, x cannot be negative, so we take x = 3.
Now, we can substitute the value of x = 3 back into the equations to find the individual values for each set:
- 10 paintings are neither T nor B.
- x(x-2) = 3(3-2) = 3 paintings are T but not B.
- x = 3 paintings are both T and B.
- 2x+8 = 2(3) + 8 = 14 paintings are B.
To find the probability of a painting chosen at random being British, given that it is from the 20th century, we use the formula:
Probability = Number of favorable outcomes / Total number of possible outcomes
In this case, the number of favorable outcomes is the number of British paintings (14), and the total number of possible outcomes is the total number of paintings from the 20th century (3 + 3 + 14 = 20).
So, the probability of a painting chosen at random being British, given that it is from the 20th century, is:
Probability = 14 / 20 = 7/10 = 0.7
Therefore, the answer to the question is 0.7 or 7/10, not 1/5.