Questions: Consider the following Venn diagram, where P(E1) = P(E2) = P(E3) = 1/5, P(E4) = P(E5) = 1/20, P(E6) = 1/10, and P(E7) = 1/5. Complete a through h. Click the icon to view the Venn diagram. a. Find P(A). P(A) = 1 (Type an integer or a simplified fraction.)

Consider the following Venn diagram, where P(E1) = P(E2) = P(E3) = 1/5, P(E4) = P(E5) = 1/20, P(E6) = 1/10, and P(E7) = 1/5. Complete a through h. Click the icon to view the Venn diagram. a. Find P(A). P(A) = 1 (Type an integer or a simplified fraction.)
Transcript text: Consider the following Venn diagram, where $P\left(E_{1}\right)=P\left(E_{2}\right)=P\left(E_{3}\right)=\frac{1}{5}, P\left(E_{4}\right)=P\left(E_{5}\right)=\frac{1}{20}, P\left(E_{6}\right)=\frac{1}{10}$, and $P\left(E_{7}\right)=\frac{1}{5}$, Complete a through $h$ Click the icon to view the Venn diagram. a. Find $P(A)$. $P(A)=1$ (Type an integer or a simplified fraction.) Figure
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Solution

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Solution Steps

Step 1: Identify the given probabilities

The problem provides the following probabilities:

  • \( P(E_1) = P(E_2) = P(E_3) = P(E_7) = \frac{1}{5} \)
  • \( P(E_4) = \frac{1}{20} \)
  • \( P(E_5) = \frac{1}{20} \)
  • \( P(E_6) = \frac{1}{10} \)
Step 2: Understand the Venn Diagram

The Venn diagram shows two overlapping circles, A and B, with the following regions:

  • \( E_1 \) and \( E_2 \) are in circle A.
  • \( E_2 \) and \( E_3 \) are in circle B.
  • \( E_4 \), \( E_5 \), \( E_6 \), and \( E_7 \) are outside both circles.
Step 3: Calculate \( P(A) \)

To find \( P(A) \), sum the probabilities of the events within circle A:

  • \( P(A) = P(E_1) + P(E_2) \)

Given:

  • \( P(E_1) = \frac{1}{5} \)
  • \( P(E_2) = \frac{1}{5} \)

So, \[ P(A) = \frac{1}{5} + \frac{1}{5} = \frac{2}{5} \]

Final Answer

\[ P(A) = \frac{2}{5} \]

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