Probabilities

Probability is how likely a specific outcome is to happen, when we don’t know what the result of some action will be. We express probability as a fraction, percentage, or decimal.  We generally first need the fraction form to calculate the decimal or percentage.  

An example:

If you have a bag with 4 red marbles, 6 black marbles, and 5 white marbles, what are the odds that you will draw a red marble if you remove one without looking?

A) 4
B) \(\frac46\)
C) \(\frac4{15}\)
D) \(\frac{11}{15}\)

Strategy:

  1. The outcome we are looking for always goes on top (numerator) .
  2. The total possible outcomes are always on the bottom (denominator).

In our example, the thing we are looking for is red marbles.  There are 4 of them.  That goes on top.  We could draw any of the marbles, so 4 + 6 + 5 = 15 total possible outcomes.  That goes on the bottom, and C) is our answer.

So far, pretty easy.  Unfortunately, the SAT tries to make things much trickier for us.  They will use two-way tables, and specific denominators other than the basic total.

Two-way tables present counts of things using two variables.  Here’s an easy example:

Car Accidents by Age and Sex

MaleFemale
16 years old14389
18 years old11172

To get values, we look at the intersection of the two variables.  If we were asked how many 16-year-old females were in car accidents, the count at the intersection is 89.

Here’s a little more complex example, approach the table with the same intersection method.

Population of Moldova by Region and Primary Language, in 1,000s

EnglishRussianGagauzMoldovanUkrainianFrenchTotal
East128134134152144140832
West82708890103110543
South172223242823137
North2172452763012802521571
Total4444715215675555253083

Consider the following problems that might appear along with the table above:

  1. The above table summarizes the results of a survey of Moldovans by region and primary language.  If one Moldovan is chosen at random, what is the probability that they are a Gagauz speaker from the East?
  2. The above table summarizes the results of a survey of Moldovans by region and primary language.  What is the probability a Russian speaker is from the South?
  3. The above table summarizes the results of a survey of Moldovans by region and primary language.  If one of the Moldovans from the East is chosen at random, what is the probability that they do not speak English or French?
  4. The above table summarizes the results of a survey of Moldovans by region and primary language.  What proportion of people who speak Russian and Ukrainian live in the North?

We approach all these the same way, first find the thing we are looking for, then find the specific total we are choosing from.

In the first example, we need to find Gagauz speakers from the East (134).  Then, we identify that it is out of all Moldovans, so we put that over 3083.  So our probability is \(\frac{134}{3083}\).

In the second, we need to find Russian speakers from the South (22).  Notice, though, that we’re not looking at all Moldovans, only the Russian speakers.  So the probability is \(\frac{22}{471}\).

In the third, we have to find people from the East who do not speak English or French, so we have to do some addition to find that it’s 564.  Again, we have restricted our total pool to be only people from the East, so the denominator is 832. So our probability is \(\frac{564}{832}\).

For the fourth example, first we look for all the Russians and Ukrainians (as this is our total we’re choosing from): \(471+555=1,026\). The numerator is those who speak those languages and who live in the north: \(245+280=525\). Our probability is ready! \(\frac{525}{1,026}\).