break it down into cases

at most one day of rain in 3 days to me means we could have:

norain norain norain ----> (1/3)^3 = 1/27
rain norain norain -------> 2/3(1/3^2 = 2/27
norain rain norain -------> same as above = 2/27
norain norain rain -------> same as above = 2/27

since there is an OR between these probabilities and there are no overlaps we simply add these
so 1/27 + 3(2/27) = 7/27

The probability of rain on any given day is 2/3. What is the probablity of at most one day of rain during the next three days...i don't get this problem plz explain the steps :)

P(A and B) = P(A) * P(B)

2/3 = rain
1/3 = no rain

2/3 * 1/3 = 2/9

To calculate the probability of at most one day of rain during the next three days, we need to consider two possible cases: 0 days of rain and 1 day of rain.

Let's break down the probability calculation for each case:

Case 1: 0 days of rain
In this case, for each of the three days, there is no rain. The probability of no rain on any given day is 1 - 2/3 = 1/3. Since this case has to occur on all three days, we multiply the probability together:
(1/3) x (1/3) x (1/3) = 1/27

Case 2: 1 day of rain
In this case, we have 1 day of rain and 2 days with no rain. The rain can occur on any of the three days, so we need to consider the different possibilities.

a) Rain on the first day:
The probability of rain on the first day is 2/3. For the remaining two days, there is no rain, which has a probability of 1/3. So the probability in this case is:
(2/3) x (1/3) x (1/3) = 2/27

b) Rain on the second day:
The probability of rain on the second day is also 2/3. For the first and third days, there is no rain, so we multiply the probabilities:
(1/3) x (2/3) x (1/3) = 2/27

c) Rain on the third day:
The probability of rain on the third day is still 2/3. For the first two days, there is no rain, so we multiply the probabilities:
(1/3) x (1/3) x (2/3) = 2/27

Since these three cases are mutually exclusive (meaning they cannot occur at the same time), we can add up the probabilities:
(2/27) + (2/27) + (2/27) = 6/27 = 2/9

Finally, to calculate the probability of at most one day of rain during the next three days, we add the probabilities from Case 1 and Case 2:
(1/27) + (2/9) = 7/27

Therefore, the probability of at most one day of rain during the next three days is 7/27.