Respuesta :
Note that the two events are mutually exclusive. If the question is answered correctly on the first try, there's no need to give it another attempt. So [tex]\mathbb P(A\cap B)=0[/tex].
We're given that [tex]P(A)=0.2[/tex] and [tex]P(B\mid A^C)=0.5[/tex]. From the first probability, we know that [tex]P(A^C)=1-0.2=0.8[/tex]. By definition of conditional probability,
[tex]\mathbb P(B\mid A^C)=\dfrac{\mathbb P(B\cap A^C)}{\mathbb P(A^C)}[/tex]
[tex]\implies\mathbb P(B\cap A^C)=0.5\cdot0.8=0.4[/tex]
We're interested in the probability of either [tex]A[/tex] or [tex]B[/tex] occurring, i.e. [tex]\mathbb P(A\cup B)[/tex]. Apply the inclusion-exclusion principle, which says
[tex]\mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B)-\mathbb P(A\cap B)[/tex]
We know the probability of intersection is 0, and we know [tex]\mathbb P(A)[/tex]. Meanwhile, by the law of total probability, we have
[tex]\mathbb P(B)=\mathbb P(B\cap A)+\mathbb P(B\cap A^C)=\mathbb P(B\cap A^C)[/tex]
so we end up with
[tex]\mathbb P(A\cup B)=0.2+0.4=0.6[/tex]
We're given that [tex]P(A)=0.2[/tex] and [tex]P(B\mid A^C)=0.5[/tex]. From the first probability, we know that [tex]P(A^C)=1-0.2=0.8[/tex]. By definition of conditional probability,
[tex]\mathbb P(B\mid A^C)=\dfrac{\mathbb P(B\cap A^C)}{\mathbb P(A^C)}[/tex]
[tex]\implies\mathbb P(B\cap A^C)=0.5\cdot0.8=0.4[/tex]
We're interested in the probability of either [tex]A[/tex] or [tex]B[/tex] occurring, i.e. [tex]\mathbb P(A\cup B)[/tex]. Apply the inclusion-exclusion principle, which says
[tex]\mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B)-\mathbb P(A\cap B)[/tex]
We know the probability of intersection is 0, and we know [tex]\mathbb P(A)[/tex]. Meanwhile, by the law of total probability, we have
[tex]\mathbb P(B)=\mathbb P(B\cap A)+\mathbb P(B\cap A^C)=\mathbb P(B\cap A^C)[/tex]
so we end up with
[tex]\mathbb P(A\cup B)=0.2+0.4=0.6[/tex]
The probability that the student answers the question on the first or second try is 0.60.
Given
P(A) = 0.2
[tex]\rm P(B|A^c)=0.5[/tex].
What is conditional probability?
The conditional probability of an event is when the probability of one event depends on the probability of occurrence of the other event.
When two events are mutually exclusive.
Then,
[tex]\rm P(A\cap B)=0[/tex]
The first probability is;
[tex]\rm P(A^C)=1-0.2=0.8[/tex]
Therefore,
The probability that the student answers the question on the first or second try is;
[tex]\rm P(A\cup B)= P(A) +P(B)-P(A \cap B)\\\\ P(A\cup B)= P(A) +P(B|A^C) \times P(A^C)-P(A \cap B)\\\\ P(A\cup B)= 0.2+0.5 \times 0.8 -0\\\\P(A\cup B)=0.2+0.40\\\\ P(A\cup B)=0.60[/tex]
Hence, the probability that the student answers the question on the first or second try is 0.60.
To know more about Conditional probability click the link given below.
https://brainly.com/question/10739947
