The difference in the probability between weather and coin flips.

[fallingknives]

I am reading this book to try to learn.

In the first few pages I already have a question. I will post the page.






Here is my question. Why can't this be flipped around? Instead of the probability it won't rain, switch it to the probability it will. The probability that it does rain on Saturday is 50, same with Sunday. .5 x .5 =.25. That means there is a 25% chance it will rain and a 75% chance it won't.

Also, with a coin flip example. If i say on saturday we are gonna flip a coin once and it is 50% it will be heads, and sunday we are gonna flip a coin once and 50% chance of heads. What is the chance of tails coming up over the weekend? Isn't it still 50%?

 

[Steven G]

You need to be more process with The probability that it does rain on Saturday is 50, same with Sunday. .5 x .5 =.25. That means there is a 25% chance it will rain and a 75% chance it won't.

What exactly do you mean by .5 x .5 = .25, ie what probability is this? You first need to know this answer to begin seeing where you are wrong.

 

[fallingknives]

You need to be more process with The probability that it does rain on Saturday is 50, same with Sunday. .5 x .5 =.25. That means there is a 25% chance it will rain and a 75% chance it won't.

What exactly do you mean by .5 x .5 = .25, ie what probability is this? You first need to know this answer to begin seeing where you are wrong.

Did you read the passage from the picture I posted?

The chance it will rain on Saturday is 50%. The chance it will rain on Sunday is 50% What are the chances it will rain over the weekend? According to the book, we multiply the probability it WON'T rain on Saturday by the probability it WON'T rain on Sunday . .5 x .5 =25%. So there is a 25 chance it WON'T rain over the weekend. My question is, why not use WILL rain, instead of WON'T rain?

If we take the probability it will rain on Saturday and multiply it by probability it will rain on sunday, it is the same thing. There is a 25% chance it WILL rain, therefore a 75% chance it won't.

 

[Steven G]

I am still going to ask the same question,
You said If we take the probability it will rain on Saturday and multiply it by probability it will rain on sunday, it is the same thing.
Please state in your own words what you are taking the probability of to get .5 x .5.
Your answer can be in the form p ( ?????) = .5 x .5

You are telling us HOW to calculate the probability by saying that we take the probability it will rain on Saturday and multiply it by probability it will rain on sunday but you are not saying what you are taking the probability of!!

 

[tkhunny]

If we have the probability of rain on each day the same, with the following definitions and relationships...
a = p(rain)
b = 1-a = p(no rain)

We can calculate...
(a+b)(a+b) = a^2 + 2ab + b^2

a^2 = p(rain on both days)
b^2 = p(rain on neither day) = p(no rain over the weekend)
2ab = p(rain on only one day)
a^2 + 2ab = p(rain over the weekend) = 1-b^2

 

[tkhunny]

Of course, this still leaves the question concerning what EXACTLY is meant by a 50% chance of rain. We should also note that it is VERY unlikely that rain on two consecutive days is independent.

 

[fallingknives]

I am still going to ask the same question,
You said If we take the probability it will rain on Saturday and multiply it by probability it will rain on sunday, it is the same thing.
Please state in your own words what you are taking the probability of to get .5 x .5.
Your answer can be in the form p ( ?????) = .5 x .5

You are telling us HOW to calculate the probability by saying that we take the probability it will rain on Saturday and multiply it by probability it will rain on sunday but you are not saying what you are taking the probability of!!

Oh, the probability that it will rain over the weekend.

 

[fallingknives]

Of course, this still leaves the question concerning what EXACTLY is meant by a 50% chance of rain. We should also note that it is VERY unlikely that rain on two consecutive days is independent.

This is funny because that is exactly what this book is about. lol. Most people don't know what that means and experts in fields are not good at explaining/communicating probabilities. A lot of people think it means that it will rain for 30% of the day.

It really means that everytime this forecast for rain is made, it is true X amount of times. If there is a 70% chance of rain, it means that a forecaster has been right to predict rain in these situations 70% of the time.

 

[Dr.Peterson]

The chance it will rain on Saturday is 50%. The chance it will rain on Sunday is 50% What are the chances it will rain over the weekend? According to the book, we multiply the probability it WON'T rain on Saturday by the probability it WON'T rain on Sunday . .5 x .5 =25%. So there is a 25 chance it WON'T rain over the weekend. My question is, why not use WILL rain, instead of WON'T rain?

If we take the probability it will rain on Saturday and multiply it by probability it will rain on Sunday, it is the same thing. There is a 25% chance it WILL rain, therefore a 75% chance it won't.

Assuming that the probabilities are for whether there will be any rain on Saturday, and whether there will be any rain on Sunday, and that (probably not true) these are independent, their calculation gives the probability that it will not rain on Saturday and will not rain on Sunday as .5*.5 = .25; this is because we multiply independent probabilities to find the probability that both will happen ("and"). So this is the probability that it will not rain over the weekend; the probability that it will is 1 - .25 = .75.

Your calculation, multiplying the probabilities that it will rain on Saturday and that it will rain on Sunday, gives the probability that it will rain on both days. That's not what you are trying to calculate!

The fact is that it will rain on neither day .25 of the time, on both days .25 of the time, and on exactly one day .5 of the time. The probability that it will rain on at least one day is .25 + .5 = .75, providing another way to get the same answer.

The overall method, P(at least one of A and B) = 1 - P(not A)*P(not B), is a standard way to find a probability of "at least one" as the complement of the probability of "neither".

 

[Steven G]

I most definitely want to repeat what DR Peterson said:

Your calculation, multiplying the probabilities that it will rain on Saturday and that it will rain on Sunday, gives the probability that it will rain on both days. That's not what you are trying to calculate!

 

[fallingknives]

Assuming that the probabilities are for whether there will be any rain on Saturday, and whether there will be any rain on Sunday, and that (probably not true) these are independent, their calculation gives the probability that it will not rain on Saturday and will not rain on Sunday as .5*.5 = .25; this is because we multiply independent probabilities to find the probability that both will happen ("and"). So this is the probability that it will not rain over the weekend; the probability that it will is 1 - .25 = .75.

Your calculation, multiplying the probabilities that it will rain on Saturday and that it will rain on Sunday, gives the probability that it will rain on both days. That's not what you are trying to calculate!

The fact is that it will rain on neither day .25 of the time, on both days .25 of the time, and on exactly one day .5 of the time. The probability that it will rain on at least one day is .25 + .5 = .75, providing another way to get the same answer.

The overall method, P(at least one of A and B) = 1 - P(not A)*P(not B), is a standard way to find a probability of "at least one" as the complement of the probability of "neither".

Thank you. This clears it up. I didn't notice it I was calculating probability for raining on both days. The problem makes sense now. I missed the "and" out of my own ignorance. I haven't taken a math course in a long time. Thanks for the lesson.

 

[Steven G]

Thank you. This clears it up. I didn't notice it I was calculating probability for raining on both days. The problem makes sense now. I missed the "and" out of my own ignorance. I haven't taken a math course in a long time. Thanks for the lesson.

This is why you need to try to answer our leading questions. It is good that you now understand the problem now but figuring out for yourself, with our guidance, makes you even tougher the next time you have to reason through something. Good luck!