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If
P(B)=0.55,P(A∣B)=0.80,P(B^('))=0.45, and
P(A∣B^('))=0.40, find
P(B∣A)

If $P(B)=0.55, P(A \mid B)=0.80, P\left(B^{\prime}\right)=0.45$ , and $P\left(A \mid B^{\prime}\right)=0.40$ , find $P(B \mid A)$

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Evaluate expression when a complex numbers and a variable term is given

Full solution

Q. If

P(B)=0.55, P(A \mid B)=0.80, P\left(B^{\prime}\right)=0.45

, and

P\left(A \mid B^{\prime}\right)=0.40

, find

P(B \mid A)

Rephrase Problem: First, let's rephrase the ＂Find the conditional probability $P(B|A)$ .＂
Apply Bayes' Theorem: To find $P(B|A)$ , we will use Bayes' theorem, which states that $P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$ .
Calculate $P(A)$ : We already have $P(A|B) = 0.80$ and $P(B) = 0.55$ . We need to find $P(A)$ , which is the total probability of $A$ occurring.
Use Law of Total Probability: To find $P(A)$ , we use the law of total probability: $P(A) = P(A|B) \cdot P(B) + P(A|B') \cdot P(B')$ .
Calculate $P(A)$ : We have $P(A|B) = 0.80$ , $P(B) = 0.55$ , $P(A|B') = 0.40$ , and $P(B') = 0.45$ . Let's calculate $P(A)$ . $\newline$ $P(A) = (0.80 \times 0.55) + (0.40 \times 0.45)$ $\newline$ $P(A) = 0.44 + 0.18$ $\newline$ $P(A) = 0.62$
Calculate $P(B|A)$ : Now we can calculate $P(B|A)$ using Bayes' theorem. $\newline$ $P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$ $\newline$ $P(B|A) = \frac{(0.80 \cdot 0.55)}{0.62}$ $\newline$ $P(B|A) = \frac{0.44}{0.62}$
Simplify Fraction: Finally, we simplify the fraction to get the value of $P(B|A)$ . $P(B|A) = \frac{0.44}{0.62} \approx 0.7097$

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