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Answer :
Let's solve the problem step-by-step. We are given the following information:
- [tex]\( n(\vec{E}) = 32 \)[/tex], which is the number of elements in set [tex]\( E \)[/tex].
- [tex]\( n(\vec{F}) = 40 \)[/tex], which is the number of elements in set [tex]\( F \)[/tex].
- [tex]\( n(\tilde{E} \cap \vec{F}) = 20 \)[/tex], which is the number of elements in the intersection of the complement of [tex]\( E \)[/tex] and [tex]\( F \)[/tex].
We need to find [tex]\( P(I \mid E) \)[/tex], which is interpreted as the probability of event [tex]\( I \)[/tex] given that we are considering the set [tex]\( E \)[/tex]. However, in this context, let's focus on finding the overlap or intersection of [tex]\( E \)[/tex] and [tex]\( F \)[/tex].
1. Calculate [tex]\( n(E \cap F) \)[/tex]:
The intersection [tex]\( n(E \cap F) \)[/tex] can be found using the formula:
[tex]\[
n(E \cap F) = n(F) - n(\tilde{E} \cap F)
\][/tex]
Substituting the values we have:
[tex]\[
n(E \cap F) = 40 - 20 = 20
\][/tex]
2. Calculate [tex]\( P(I \mid E) \)[/tex]:
This represents the probability of the intersection of [tex]\( E \)[/tex] and [tex]\( F \)[/tex] out of the total elements in [tex]\( E \)[/tex]. Hence:
[tex]\[
P(I \mid E) = \frac{n(E \cap F)}{n(E)}
\][/tex]
Substituting the known values:
[tex]\[
P(I \mid E) = \frac{20}{32}
\][/tex]
Simplifying the fraction gives:
[tex]\[
P(I \mid E) = \frac{5}{8}
\][/tex]
Therefore, the correct choice for [tex]\( P(I \mid E) \)[/tex] is [tex]\(\frac{5}{8}\)[/tex].
- [tex]\( n(\vec{E}) = 32 \)[/tex], which is the number of elements in set [tex]\( E \)[/tex].
- [tex]\( n(\vec{F}) = 40 \)[/tex], which is the number of elements in set [tex]\( F \)[/tex].
- [tex]\( n(\tilde{E} \cap \vec{F}) = 20 \)[/tex], which is the number of elements in the intersection of the complement of [tex]\( E \)[/tex] and [tex]\( F \)[/tex].
We need to find [tex]\( P(I \mid E) \)[/tex], which is interpreted as the probability of event [tex]\( I \)[/tex] given that we are considering the set [tex]\( E \)[/tex]. However, in this context, let's focus on finding the overlap or intersection of [tex]\( E \)[/tex] and [tex]\( F \)[/tex].
1. Calculate [tex]\( n(E \cap F) \)[/tex]:
The intersection [tex]\( n(E \cap F) \)[/tex] can be found using the formula:
[tex]\[
n(E \cap F) = n(F) - n(\tilde{E} \cap F)
\][/tex]
Substituting the values we have:
[tex]\[
n(E \cap F) = 40 - 20 = 20
\][/tex]
2. Calculate [tex]\( P(I \mid E) \)[/tex]:
This represents the probability of the intersection of [tex]\( E \)[/tex] and [tex]\( F \)[/tex] out of the total elements in [tex]\( E \)[/tex]. Hence:
[tex]\[
P(I \mid E) = \frac{n(E \cap F)}{n(E)}
\][/tex]
Substituting the known values:
[tex]\[
P(I \mid E) = \frac{20}{32}
\][/tex]
Simplifying the fraction gives:
[tex]\[
P(I \mid E) = \frac{5}{8}
\][/tex]
Therefore, the correct choice for [tex]\( P(I \mid E) \)[/tex] is [tex]\(\frac{5}{8}\)[/tex].
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