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There are two known issues with a certain model of new car.

The first issue, A, occurs with a probability of [tex]\( P(A) = 0.1 \)[/tex].

B is another known issue with the car. If it is known that either event occurs with a probability of [tex]\( P(A \text{ OR } B) = 0.93 \)[/tex], and that both events occur with a probability of [tex]\( P(A \text{ AND } B) = 0.07 \)[/tex], calculate [tex]\( P(B) \)[/tex].



Answer :

GinnyAnswer
Certainly! Let's break down the calculation step-by-step.

We need to find the probability of issue [tex]$B$[/tex] occurring, denoted as [tex]$P(B)$[/tex].

We know:
- The probability of issue [tex]$A$[/tex] occurring, [tex]\( P(A) = 0.1 \)[/tex].
- The probability of either issue [tex]$A$[/tex] or issue [tex]$B$[/tex] occurring (or both), [tex]\( P(A \text{ OR } B) = 0.93 \)[/tex].
- The probability of both issues [tex]$A$[/tex] and [tex]$B$[/tex] occurring together, [tex]\( P(A \text{ AND } B) = 0.07 \)[/tex].

We can use the formula for the union of two events to find [tex]\( P(B) \)[/tex]:

[tex]\[ P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B) \][/tex]

Now we will rearrange this formula to solve for [tex]\( P(B) \)[/tex]:

[tex]\[ P(B) = P(A \text{ OR } B) - P(A) + P(A \text{ AND } B) \][/tex]

Substituting in the known values:

[tex]\[ P(B) = 0.93 - 0.1 + 0.07 \][/tex]

Then, we perform the subtraction and addition:

[tex]\[ P(B) = 0.93 - 0.1 = 0.83 \][/tex]
[tex]\[ P(B) = 0.83 + 0.07 = 0.90 \][/tex]

Therefore, the probability of issue [tex]$B$[/tex] occurring is:

[tex]\[ P(B) = 0.90 \][/tex]

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