Question

1. using the venn diagram shown, find: probability = \\(\frac{28}{43}\\) 5 great job! 6c \\(p(\text{not } a)\\) probability = \\(\frac{29}{43}\\) 5 nice work! 6d \\(p(\text{not } b)\\) probability = enter your next step here

Explanation:

Step1: Recall the total number of elements

From previous parts, we know the total number of elements (the denominator of the probability) is 43.

Step2: Find the number of elements in B (from Venn diagram logic, though we can infer from complementary probabilities)

First, we know that \( P(A) = 1 - P(\text{not } A) = 1 - \frac{29}{43} = \frac{14}{43} \). But for \( P(\text{not } B) \), we can also use the fact that \( P(\text{not } B) = 1 - P(B) \). Alternatively, from the total and the number of elements not in B. Wait, actually, since the total is 43, and if we consider the previous probability \( P(A) \) was related, but more directly, let's think: in a Venn diagram, the total is 43. Let's assume that the number of elements in B: wait, maybe from the first part, the probability was \( \frac{28}{43} \), maybe that was \( P(A \text{ or something}) \), but actually, for \( P(\text{not } B) \), we know that \( P(\text{not } B) = 1 - P(B) \). But also, from the total, if we look at \( P(\text{not } A) = \frac{29}{43} \), which is the number of elements not in A. But for \( P(\text{not } B) \), let's think of the total number of elements not in B. Wait, maybe the total number of elements is 43. Let's suppose that the number of elements in B is \( 43 - \text{number of elements not in B} \). Alternatively, from the first probability given, \( \frac{28}{43} \), maybe that was \( P(A) \)? Wait, no, \( P(\text{not } A) = \frac{29}{43} \), so \( P(A) = 1 - \frac{29}{43} = \frac{14}{43} \). Wait, maybe the first probability was \( P(A \cup B) \) or something, but actually, for \( P(\text{not } B) \), let's use the formula \( P(\text{not } B) = 1 - P(B) \). But we can also find the number of elements not in B. Let's think: the total is 43. If we look at the previous answer for \( P(\text{not } A) = \frac{29}{43} \), which is the number of elements not in A. But for \( P(\text{not } B) \), let's assume that the number of elements in B is \( 43 - \text{number of elements not in B} \). Wait, maybe from the first probability, \( \frac{28}{43} \) was \( P(B) \)? Wait, no, let's check: if \( P(\text{not } B) = 1 - P(B) \), and if we can find \( P(B) \). Wait, maybe the first probability given was \( P(A) = \frac{14}{43} \) (since \( P(\text{not } A) = \frac{29}{43} \)), but no, that doesn't make sense. Wait, actually, in a Venn diagram, the total number of elements is the sum of all regions. Let's suppose that the total number of elements is 43 (from the denominators). So to find \( P(\text{not } B) \), we need the number of elements not in B divided by 43. Let's think: if \( P(\text{not } A) = \frac{29}{43} \), that means there are 29 elements not in A. Then the number of elements in A is \( 43 - 29 = 14 \). Now, if we look at the first probability, \( \frac{28}{43} \), maybe that was \( P(B) \)? Wait, no, \( P(B) \) would be the number of elements in B divided by 43. Wait, maybe the number of elements in B is 28? No, because \( P(\text{not } B) = 1 - \frac{28}{43} \)? Wait, no, that would be \( \frac{15}{43} \), but that doesn't match. Wait, no, wait the first probability was \( \frac{28}{43} \), maybe that was \( P(A \cap B) \) or something else. Wait, no, let's re-express:

We know that the total number of elements (the sample space) is 43 (since all probabilities have denominator 43).

To find \( P(\text{not } B) \), we use the formula:

\( P(\text{not } B) = 1 - P(B) \)

But we need to find \( P(B) \). Wait, maybe from the first part, the probability was \( P(A) = \frac{14}{43} \) (since \( P(\text{not } A) = \frac{29}{43} \)), but that's not helpful. Wait, no, maybe the number of elements in B is \( 43 - \text{number of elements not in B} \). Wait, let's think differently. Let's assume that in the Venn diagram, the number of elements not in B is equal to the number of elements not in B. Wait, maybe the first probability given was \( P(A) = \frac{14}{43} \) (since \( 43 - 29 = 14 \)), and the other probability was \( \frac{28}{43} \), maybe that was \( P(B) \)? Wait, no, \( \frac{28}{43} \) is more than half. Wait, maybe the correct approach is:

Total number of elements: 43 (from the denominators of the probabilities).

To find \( P(\text{not } B) \), we need the number of elements not in B divided by 43.

From the previous part, \( P(\text{not } A) = \frac{29}{43} \), which is the number of elements not in A. But for \( P(\text{not } B) \), let's suppose that the number of elements in B is 28? Wait, no, the first probability was \( \frac{28}{43} \), maybe that was \( P(B) \). Then \( P(\text{not } B) = 1 - \frac{28}{43} = \frac{15}{43} \)? Wait, no, that can't be. Wait, no, the first probability was maybe \( P(A) \), but \( P(\text{not } A) = \frac{29}{43} \), so \( P(A) = \frac{14}{43} \). Then if \( P(B) \) is, say, \( \frac{28}{43} \), then \( P(\text{not } B) = 1 - \frac{28}{43} = \frac{15}{43} \). But that doesn't seem right. Wait, maybe I made a mistake. Wait, let's check the numbers again.

Wait, the total number of elements is 43.

For \( P(\text{not } A) = \frac{29}{43} \), so number of elements not in A is 29. Therefore, number of elements in A is \( 43 - 29 = 14 \).

Now, for \( P(\text{not } B) \), we need the number of elements not in B. Let's assume that the number of elements in B is, say, 28 (from the first probability \( \frac{28}{43} \)). Then number of elements not in B is \( 43 - 28 = 15 \). Therefore, \( P(\text{not } B) = \frac{15}{43} \)? Wait, no, that contradicts. Wait, no, maybe the first probability was \( P(A \cup B) \) or something else. Wait, no, the problem is about Venn diagrams, so the total number of elements is the sum of all regions: only A, only B, both A and B, and neither.

Let me denote:

- Only A: \( a \)
- Only B: \( b \)
- Both A and B: \( c \)
- Neither: \( d \)

Total: \( a + b + c + d = 43 \)

\( P(\text{not } A) = \frac{b + d}{43} = \frac{29}{43} \), so \( b + d = 29 \)

\( P(A) = \frac{a + c}{43} = 1 - \frac{29}{43} = \frac{14}{43} \), so \( a + c = 14 \)

Now, for \( P(\text{not } B) \), that is \( \frac{a + d}{43} \)

We need to find \( a + d \)

From the first probability, maybe \( P(A \text{ or B}) = \frac{28}{43} \), which is \( \frac{a + b + c}{43} = \frac{28}{43} \), so \( a + b + c = 28 \)

Then, since \( a + c = 14 \), then \( b = 28 - 14 = 14 \)

Then, from \( b + d = 29 \), \( d = 29 - 14 = 15 \)

Then, \( a + d = (a + c) - c + d = 14 - c + 15 = 29 - c \). Wait, no, better: \( a + d = (a + c) + d - c = 14 + d - c \). But we know \( a + c = 14 \) and \( b + d = 29 \), \( b = 14 \), so \( d = 15 \). Also, \( a + b + c + d = 43 \), so \( 14 + 14 + c + 15 = 43 \), so \( c + 43 = 43 \), so \( c = 0 \). Wait, that can't be. So \( c = 0 \), then \( a = 14 - 0 = 14 \), \( b = 14 \), \( d = 15 \)

Then, \( P(\text{not } B) = \frac{a + d}{43} = \frac{14 + 15}{43} = \frac{29}{43} \)? No, that's not right. Wait, I'm confused. Wait, maybe the first probability was \( P(B) = \frac{28}{43} \), so \( b + c = 28 \). Then, \( a + d = 43 - 28 = 15 \), so \( P(\text{not } B) = \frac{15}{43} \). But then \( P(\text{not } A) = \frac{b + d}{43} = \frac{29}{43} \), so \( b + d = 29 \). If \( b + c = 28 \) and \( b + d = 29 \), then \( d - c = 1 \). Also, \( a + c + b + d = 43 \), so \( a + (b + c) + d = 43 \), so \( a + 28 + d = 43 \), so \( a + d = 15 \), which is \( P(\text{not } B) \). Then, from \( b + d = 29 \) and \( b + c = 28 \), we get \( d = c + 1 \). Then, \( a + d = 15 \), so \( a + c + 1 = 15 \), so \( a + c = 14 \), which is \( P(A) = \frac{14}{43} \), which matches \( P(\text{not } A) = \frac{29}{43} \). So that works. Therefore, \( P(\text{not } B) = \frac{15}{43} \)? Wait, no, \( a + d = 15 \), so \( P(\text{not } B) = \frac{15}{43} \). But that seems conflicting. Wait, maybe I made a mistake in the Venn diagram regions. Alternatively, maybe the first probability given was \( P(B) = \frac{28}{43} \), so \( P(\text{not } B) = 1 - \frac{28}{43} = \frac{15}{43} \). Yes, that makes sense. So the answer is \( \frac{15}{43} \).

Wait, let's verify:

Total elements: 43.

If \( P(B) = \frac{28}{43} \), then number of elements in B is 28.

Therefore, number of elements not in B is \( 43 - 28 = 15 \).

Thus, \( P(\text{not } B) = \frac{15}{43} \).

Yes, that makes sense. So the steps are:

Step1: Recall the total number of elements (denominator) is 43.

Step2: Find \( P(B) \) from previous (or infer from the first probability given, \( \frac{28}{43} \)).

Step3: Calculate \( P(\text{not } B) = 1 - P(B) = 1 - \frac{28}{43} = \frac{43 - 28}{43} = \frac{15}{43} \).

Answer:

\(\frac{15}{43}\)