The weather channel has predicted that there is a 70% chance of rain today and 20% chance of rain tomorrow. A. What is the probability it wont rain tomorrow? B. What is the probability it wont rain tomorrow given that it rained today?
a. wont rain tomorrow? .8
B cannot be answered, there is no indication that rain on successive days are independent events.
To calculate the probability, we need to use the concept of complement. The complement of an event is the opposite outcome. For example, if the event is "rain," the complement would be "no rain."
A. To calculate the probability that it won't rain tomorrow, we subtract the probability of rain from 100%, since the sum of these two probabilities should equal 100%. Therefore,
Probability of no rain tomorrow = 100% - Probability of rain tomorrow
= 100% - 20%
= 80%
So, there is an 80% chance that it won't rain tomorrow.
B. To calculate the probability that it won't rain tomorrow given that it rained today, we need to understand the concept of conditional probability. Conditional probability refers to the probability of an event occurring given that another event has already occurred.
In this case, we want to find the probability that it won't rain tomorrow, given the information that it rained today. This can be denoted as P(no rain tomorrow | rain today). We can calculate this using Bayes' theorem:
P(no rain tomorrow | rain today) = P(no rain tomorrow and rain today) / P(rain today)
Let's start by calculating P(rain today). The weather channel predicted a 70% chance of rain today, therefore,
P(rain today) = 70%
Next, we want to find P(no rain tomorrow and rain today). These two events are independent of each other. So, we can multiply the probabilities of individual events to find the joint probability:
P(no rain tomorrow and rain today) = P(no rain tomorrow) * P(rain today)
= (100% - 20%) * 70%
= 80% * 70%
= 56%
Putting it all together:
P(no rain tomorrow | rain today) = P(no rain tomorrow and rain today) / P(rain today)
= 56% / 70%
≈ 0.8
So, given that it rained today, there is approximately an 80% chance that it won't rain tomorrow.