Answer :
To solve the problem, we are given:
- [tex]\( n(E) = 32 \)[/tex]: The number of elements in set [tex]\( E \)[/tex].
- [tex]\( n(F) = 40 \)[/tex]: The number of elements in set [tex]\( F \)[/tex].
- [tex]\( n(E \cap \vec{F}) = 20 \)[/tex]: The number of elements in set [tex]\( E \)[/tex] but not in set [tex]\( F \)[/tex].
We need to find the probability [tex]\( P(\bar{M} \bar{E}) \)[/tex], where [tex]\( \bar{M} \)[/tex] denotes some event related to [tex]\( F \)[/tex] (complementary or otherwise) and relevant to this question as [tex]\( \bar{E} \)[/tex] suggests complement to [tex]\( E \)[/tex].
Here's the process to find the answer:
1. Determine the number of elements in the intersection of sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex], denoted as [tex]\( n(E \cap F) \)[/tex].
- We know [tex]\( n(E \cap \vec{F}) = 20 \)[/tex], this is the part of [tex]\( E \)[/tex] that is not in [tex]\( F \)[/tex].
- So, [tex]\( n(E \cap F) = n(E) - n(E \cap \vec{F}) = 32 - 20 = 12 \)[/tex].
2. Determine the probability [tex]\( P(\bar{M} \bar{E}) \)[/tex]. This can be interpreted as the probability that picks are made from outside both relevant conditions (that [tex]\( n(E \cap F) \)[/tex] represents our scenario).
- Since [tex]\( n(E \cap F) = 12 \)[/tex] gives us the overlap part relevant for this question, probability [tex]\( P \)[/tex] is based on total elements in [tex]\( F \)[/tex].
- Probability [tex]\( P(\bar{M} \bar{E}) = \frac{n(E \cap F)}{n(F)} = \frac{12}{40} = 0.3 \)[/tex].
Therefore, the probability [tex]\( P(\bar{M} \bar{E}) = 0.3 \)[/tex], which displays as a fraction is [tex]\( \frac{3}{10} \)[/tex], and that matches with option [tex]\( 0.3 \)[/tex].
This outcome confirms how these sets interact and provides the probability required by this question.
- [tex]\( n(E) = 32 \)[/tex]: The number of elements in set [tex]\( E \)[/tex].
- [tex]\( n(F) = 40 \)[/tex]: The number of elements in set [tex]\( F \)[/tex].
- [tex]\( n(E \cap \vec{F}) = 20 \)[/tex]: The number of elements in set [tex]\( E \)[/tex] but not in set [tex]\( F \)[/tex].
We need to find the probability [tex]\( P(\bar{M} \bar{E}) \)[/tex], where [tex]\( \bar{M} \)[/tex] denotes some event related to [tex]\( F \)[/tex] (complementary or otherwise) and relevant to this question as [tex]\( \bar{E} \)[/tex] suggests complement to [tex]\( E \)[/tex].
Here's the process to find the answer:
1. Determine the number of elements in the intersection of sets [tex]\( E \)[/tex] and [tex]\( F \)[/tex], denoted as [tex]\( n(E \cap F) \)[/tex].
- We know [tex]\( n(E \cap \vec{F}) = 20 \)[/tex], this is the part of [tex]\( E \)[/tex] that is not in [tex]\( F \)[/tex].
- So, [tex]\( n(E \cap F) = n(E) - n(E \cap \vec{F}) = 32 - 20 = 12 \)[/tex].
2. Determine the probability [tex]\( P(\bar{M} \bar{E}) \)[/tex]. This can be interpreted as the probability that picks are made from outside both relevant conditions (that [tex]\( n(E \cap F) \)[/tex] represents our scenario).
- Since [tex]\( n(E \cap F) = 12 \)[/tex] gives us the overlap part relevant for this question, probability [tex]\( P \)[/tex] is based on total elements in [tex]\( F \)[/tex].
- Probability [tex]\( P(\bar{M} \bar{E}) = \frac{n(E \cap F)}{n(F)} = \frac{12}{40} = 0.3 \)[/tex].
Therefore, the probability [tex]\( P(\bar{M} \bar{E}) = 0.3 \)[/tex], which displays as a fraction is [tex]\( \frac{3}{10} \)[/tex], and that matches with option [tex]\( 0.3 \)[/tex].
This outcome confirms how these sets interact and provides the probability required by this question.
