Miscellaneous problems
There are a number of real life situations where the concept of probability can be applied to get the results.
· As we've seen, to find basic probability we divide the number of favorable outcomes by the total number of possible outcomes in our sample space. If we're looking for the chance it will rain, this will be the number of days in our database that it rained divided by the total number of similar days in our database. If our meteorologist has data for 100 days with similar weather conditions (the sample space and therefore the denominator of our fraction), and on 70 of these days it rained (a favorable outcome), the probability of rain on the next similar day is 70/100 or 70%.
· In the United States, an average of 80 people are killed by lightning each year. Considering being killed by lightning to be the outcome under consideration, the sample space contains the entire population of the United States (about 250 million).
· If we assume that all the people in our sample space are equally likely to be killed by lightning (so people who never go outside have the same chance of being killed by lightning as those who stand by flagpoles in large open fields during thunderstorms), the chance of being killed by lightning in the United States is equal to 80/250 million, or a probability of about .000032%.
· Clearly, you are much more likely to die in a car accident than by being struck by lightning.
Let's say your favorite baseball player is batting 300. What does this mean?
A batting average involves calculating the probability of a player managing to hit the ball with his bat. The sample space is the total number of at-bats a player has had, not including walks. A hit is a favorable outcome. Thus if in 10 at-bats a player gets 3 hits, his or her batting average is 3/10 or 30%. For baseball stats we multiply all the percentages by 10, so a 30% probability translates to a 300 batting average.
This means that when a Major Leaguer with a batting average of 300 steps up to the plate, he has only a 30% chance of getting a hit - and since most batters hit below 300, you can see how hard it is to get a hit in the Major Leagues!
Example: The record of a weather station shows that out of the past 250 consecutive days, its weather forecast were correct 175 times. What is the probability that on a given day (i) it was correct? (ii) it was not correct?
Solution: We have,
Total number of days for which the weather forecast was made = 250
Number of days for which the forecast was correct = 175
Number of days for which the forecast was not correct = 250 – 175 = 75
Therefore,
(i) Probability that the forecast was correct on a given day
(ii) Probability that the forecast was not correct on a given day
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