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Probability in Our Lives

A basic understanding of probability makes it possible to understand everything from batting averages to the weather report or your chances of being struck by lightning! Probability is an important topic in mathematics because the probability of certain events happening - or not happening - can be important to us in the real world.

Weather forecasting

Suppose you want to go on a picnic this afternoon, and the weather report says that the chance of rain is 70%? Do you ever wonder where that 70% came from?

Forecasts like these can be calculated by the people who work for the National Weather Service when they look at all other days in their historical database that have the same weather characteristics (temperature, pressure, humidity, etc.) and determine that on 70% of similar days in the past, it rained.

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%.

Since a 50% probability means that an event is as likely to occur as not, 70%, which is greater than 50%, means that it is more likely to rain than not. But what is the probability that it won't rain? Remember that because the favorable outcomes represent all the possible ways that an event can occur, the sum of the various probabilities must equal 1 or 100%, so 100% - 70% = 30%, and the probability that it won't rain is 30%.

Flipping coins

If you want to know the probability of a coin landing heads, heads is the favorable outcome. There is only one way for a coin to land heads, so the numerator of the probability fraction is 1.

The sample space consists of the total number of ways that a coin can land. Since a coin can only land either heads or tails - 2 ways - the sample space is made up of only two possible outcomes and the denominator of the probability fraction is 2.

Thus the probability of a coin landing heads is 1/2, which is the same as saying that a coin lands heads 50% of the time.

What is the probability of the coin landing tails? We can do the same analysis as for the coin landing heads, finding a probability of 1/2, or, knowing that if a coin doesn't land heads it has to land tails, and understanding that the sum of the probabilities must equal 1, subtract: the probability of a coin landing tails must be 1 - 1/2 = 1/2.

In this case, both probabilities (a 1/2 chance of landing either heads or tails) remain true; no matter how many times you flip a coin, each time the coin is equally likely to fall heads or tails. Even if your coin has fallen heads 50 times in a row, the chance that the next toss will fall tails is still 1/2.

Batting averages

Let's say your favorite baseball player is batting 300. What does this mean?

A batting average involves calculating the probability of a player's getting a hit. 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!

Are we likely to be struck by lightning?

In the United States, an average of 80 people are killed by lightning each year. Considering being killed by lightning to be our 'favorable outcome' (not such a favorable outcome!), 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.

Combinations and permutations: the lottery

What if you want to know the probability of winning the lottery? Combination and permutation formulas are very useful for solving probability problems.

Imagine a lottery where you pick six numbers from 1-49 and for a winning number, their order matters. Here you must use the formula for permutations to figure out the size of the sample space, which consists of the number of permutations of size k that can be taken from a set of n objects:

    n_P_k = ---------
              (n - k)!

In our problem, we want to find 49_P_6, which is equal to:

    49_P_6 = ------ = 10,068,347,520

Since only one possible ordering of the six numbers can win the lottery, there is only one favorable outcome. The sample space, however, is quite large because it is equal to 49_P_6, which is roughly 10 billion. This means that the probability of winning the lottery is about 1 in 10 billion.

If we define the lottery in a slightly different way, the probability of winning greatly improves. Suppose you still pick 6 numbers from 1-49, but this time order doesn't matter. Now you can use the formula for combinations to figure out the sample space, which consists of the number of combinations of size k that can be chosen from a set of n objects:

    n_C_k = -------------
              k! (n - k)!

In our problem, we want to find 49_C_6, which is equal to:

    49_C_6 = ---------- = 13,983,816
               6! * 43!

Since there is only one combination of six numbers that will win the lottery, there is again only one favorable outcome - so your chances of choosing the winning number are quite slim. However, the sample space has shrunk considerably (by a factor of 1000) because 49_C_6 is only roughly 14 million. The probability of winning this second lottery is 1 in 14 million.

What would happen if you bought 7 million tickets?

If you picked a different combination of six numbers for each of those 7 million tickets, you'd have 7 million of the possible winning combinations and the numerator of your probability fraction would therefore be 7 million. Given the second lottery, with a sample space of 14 million possible combinations, the probability of winning the lottery is 7 million/14 million, a probability of 50%.

Thus you can see that the more tickets you buy, the better your chances of winning the lottery. However, you need to buy lots and lots of tickets before the number of tickets you hold really makes a difference. Even if you buy 100 tickets (which might cost you $100), your chances of winning would still only be 100/14 million - not even close to a 1% chance.

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