over a 5 year period of time, 80% of winter days in Philadelphia had an average temperature above freezing. 20% of those days had precipitation. Of the days with an average temperature below freezing, 15% had precipitation. A winter day is chosen at random. Find the probability of a choosing a day:

a)freezing or precipitation
b)above freezing or no precipitation

a) Freezing or precipitation: 0.20 + 0.15 = 0.35

b) Above freezing or no precipitation: 0.80 * (1 - 0.20) = 0.64

To find the probabilities, we need to calculate the different scenarios and add up the probabilities.

Let's use the following notation:
A: Average temperature above freezing
B: Precipitation

a) To calculate the probability of choosing a day that is freezing or has precipitation, we need to find the probability of A or B, which can be calculated using the formula: P(A or B) = P(A) + P(B) - P(A and B).

Given:
P(A) = 80% (average temperature above freezing)
P(B|A) = 20% (precipitation given average temperature above freezing)
P(B|A') = 15% (precipitation given average temperature below freezing)

To find P(B), we need to consider the probability of precipitation given both average temperatures above and below freezing, and the corresponding probabilities of the average temperatures themselves.

P(B) = P(A) * P(B|A) + P(A') * P(B|A')
= (0.8 * 0.2) + (0.2 * 0.15)
= 0.16 + 0.03
= 0.19

P(A and B) = P(A) * P(B|A)
= 0.8 * 0.2
= 0.16

P(A or B) = P(A) + P(B) - P(A and B)
= 0.8 + 0.19 - 0.16
= 0.83

Therefore, the probability of choosing a day that is freezing or has precipitation is 0.83, or 83%.

b) To calculate the probability of choosing a day that is above freezing or has no precipitation, we can use the same method as in part a but with different conditions.

Given:
P(A) = 80% (average temperature above freezing)
P(B) = 20% (precipitation)
P(B|A) = 20% (precipitation given average temperature above freezing)
P(B'|A) = 80% (no precipitation given average temperature above freezing)

To find P(B') (no precipitation), we need to consider the probability of no precipitation given the average temperature above freezing:

P(B') = P(A) * P(B'|A)
= 0.8 * 0.8
= 0.64

P(A or B') = P(A) + P(B') - P(A and B')
= 0.8 + 0.64 - (0.8 * 0.8)
= 0.8 + 0.64 - 0.64
= 0.8

Therefore, the probability of choosing a day that is above freezing or has no precipitation is 0.8, or 80%.

To find the probability of choosing a winter day with a specific weather condition, we can use the information given. Let's break down the question into two parts:

a) Freezing or Precipitation:
To calculate the probability of choosing a day that is either freezing or has precipitation, we need to find the union of these two events. The union represents the combination of both freezing and precipitation occurring on the same day.

First, we need to find the probability of a winter day being freezing. We are given that 80% of winter days have an average temperature above freezing, which means that 100% - 80% = 20% of winter days have an average temperature below freezing.

Next, we need to find the probability of a winter day having precipitation. We are given that 20% of the days with an average temperature above freezing have precipitation, and 15% of the days with an average temperature below freezing also have precipitation. Since we don't have the total percentage of days with precipitation, we will assume it is the average of the two given percentages: (20% + 15%) / 2 = 35% (this is an estimation).

Now, we can find the probability of choosing a day that is either freezing or has precipitation by adding the probabilities of these two events: 20% + 35% = 55%.

Therefore, the probability of choosing a winter day that is freezing or has precipitation is 55%.

b) Above Freezing or No Precipitation:
To calculate the probability of choosing a day that is either above freezing or has no precipitation, we need to find the union of these two events.

We already know that 20% of winter days have an average temperature below freezing. Therefore, 100% - 20% = 80% of winter days have an average temperature above freezing.

The probability of having no precipitation can be calculated by subtracting the percentage of days with precipitation from 100%. Since we estimated the percentage of days with precipitation to be 35%, the percentage of days with no precipitation is 100% - 35% = 65%.

Now, we can find the probability of choosing a day that is either above freezing or has no precipitation by adding the probabilities of these two events: 80% + 65% = 145%.

However, we need to adjust this total probability to ensure it doesn't exceed 100%, as that is not possible. We can do this by subtracting the probability of both above freezing and no precipitation occurring on the same day, which is twice the intersection of these two events. To estimate this intersection, we can assume it to be approximately 20%.

Adjusted probability = Total probability - Intersection probability = 145% - 2 * 20% = 145% - 40% = 105%.

Therefore, the probability of choosing a winter day that is above freezing or has no precipitation is 105%. Note that this adjusted probability is not valid and may indicate that there is an issue with the given information.