GO Classes DPP | GATE CS, DA | Probability | Bayes Theorem

Answer 1

Let,

• $F$: Event that the serve is to the Forehand. $P(F) = 0.70$

• $B$: Event that the serve is to the Backhand. $P(B) = 0.30$

• $W$: Event that she Wins the point.

Given:

• $P(W|F)$: Probability of winning given a serve to the forehand = $0.60$

• $P(W|B)$: Probability of winning given a serve to the backhand = $0.40$

 

We need to find the probability that she served to the forehand, given that we know she won the point i.e, $P(F|W)$.

 

Using Bayes Theorem,

$$P(F|W) = \frac{P(W|F) \cdot P(F)}{P(W)}$$

First, we need to find the total probability of winning, $P(W)$, which can happen in two ways (serving forehand OR serving backhand):

• $P(\text{Win via Forehand}) = P(W|F) \cdot P(F) = 0.60 \cdot 0.70 = \mathbf{0.42}$

• $P(\text{Win via Backhand}) = P(W|B) \cdot P(B) = 0.40 \cdot 0.30 = \mathbf{0.12}$

Total probability of winning:

$$P(W) = 0.42 + 0.12 = \mathbf{0.54}$$

Now, plug everything back into the main formula to find the proportion of those wins that came from forehand serves:

$$P(F|W) = \frac{0.42}{0.54}= \boxed{\mathbf{\frac{7}{9}}}$$

 

 

Comments

GO ClassesOP That's nice to add some visual picture of that solution using some AI tool ... It will very usefull to understanding the concept