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Problem statement Rain influences sprinkler usage. Rain and sprinkler influences whether grass is wet or not. What is the probability that rain gives grass wet?

Edit: The problem statement is verbatim from my question paper. I understand "rain gives grass wet" doesn't make much sense. I inferred it as "grass it wet and it rained" but I did not edit it since I'm not entirely sure.

Conditional probability values

Alternate source for image

I solved it thus:

P(G,R)=(P(G|!S,R).P(!S|R) + (P(G|S,R).P(S|R)).P(R)
        =((0.99*0.01)+(0.8*0.99))*0.2

Can someone tell me if I am right or explain why I am wrong?

Absee
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  • I think there is a translation problem here. "the probability that rain gives grass wet" doesn't make sense. What event, conditional or otherwise, are you trying to compute the probability of? – lulu May 05 '23 at 11:30
  • @lulu you're right, it doesn't. But sadly that was what was given in my question paper and my staff is miserably ill-equipped to clarify any of my doubts. Based on the solution provided in our workbook I think they're solving for "grass is wet given there was rain" and that is what I tried to solve for although I'm not sure what they mean exactly. – Absee May 05 '23 at 11:33
  • I think you meant to say "find the probability that all three events, Rain, Sprinkler, Wet Grass, happen on the same day." To do that, I would point out that you need rain, $.2$ probability, then you need the sprinkler given that it is raining, $.01$ probability, and then you need the grass to be wet given that it is raining and the sprinkler is on, $.99$ probability, so we get the product reported by your source. – lulu May 05 '23 at 11:34
  • To be clear: I just reverse engineered the given calculation and used data from the table. I ignored what you wrote about it. – lulu May 05 '23 at 11:35
  • @lulu you're right. It appears that's what they solved for. But since there's no way that's a possible takeaway from "rain gives grass wet" I've removed that solution and now am just trying to figure out if my solution is right for "grass is wet given it rains" – Absee May 05 '23 at 12:00
  • Well, that phrase is gibberish so I can't imagine a meaningful interpretation. Pretty sure they just meant the joint probability. That's clear from the official solution. I wouldn't try to assign meaning to the nonsense expression. If you have an interpretation of the phrase, I suggest you edit your post to say what it is. I'd have said that $P(G,S,R)$ meant the joint probability (all three events happen at once), but apparently you have a non-standard interpretation for that. – lulu May 05 '23 at 12:02
  • If your interpretation is $P(G,|,R)$ then we can read that off from the table. Just further condition on whether or not the sprinkler is activated. – lulu May 05 '23 at 12:05
  • @lulu So P(G|R) is ((0.990.01)+(0.80.99))? – Absee May 05 '23 at 12:16
  • Yes, that's right. – lulu May 05 '23 at 12:16
  • @lulu And P(G, R) is P(G|R)*P(R) ? – Absee May 05 '23 at 12:22
  • Let us continue this discussion in chat. – Absee May 05 '23 at 12:22
  • Yes, you are right about $P(G,R)$. That follows immediately from the definition of the conditional probability $P(G,|,R)$. – lulu May 05 '23 at 12:23

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