Data Analysis, Statistics & Probability
Probability
Probability deals with the randomness of the universe, and this inexact science vainly tries to apply numerical understanding to events that appear to happen in a noncontiguous, nonlinear environment.
What the . . . ?
Oh wait, probability on the SAT. That’s easy. Probability on the SAT is expressed as a fraction. The numerator is the number of times a certain event might occur, and the denominator is the total number of events that might occur. Here is the probability formula you need to know:
Let’s say you’re playing a game of High Card with your strange friend Rasputin. High Card is a silly game in which the two of you randomly pick cards from a 52-card deck, and the highest card wins (now that’s a Friday night of fun!). You are going to choose from the deck first, and you really want an ace. What’s the probability that you will draw an ace?
And what’s the probability that you’ll end up with a non-ace?
Here’s a more complicated example involving a more exciting Saturday night with toy cars:
9. A certain bag contains five purple toy cars, eight orange toy cars, and seven yellow toy cars. If all the toy cars are placed into a bag, what is the probability that the first car picked will be purple?
(A)
(B)
(C)
(D)
(E)
There are five ways to pick a purple toy car because there are five different purple toy cars. That’s the top number. To find the bottom number, you need to add up all the different cars, regardless of color:
5 + 8 + 7 = 20
There are a total of 20 different toy cars to choose from in all. Therefore, the probability of picking a purple toy car is:
That’s choice A.
When calculating probability, always be sure to divide by the total number of chances. It would have been tempting to leave out the number of purple toy cars in the denominator, because we used them in the numerator, which would have resulted in , or , an incorrect answer waiting for you among the distractors.
Backward Probability
Backward probability is the basic probability item asked in reverse order. Instead of finding the probability, you are looking for the total, or real number the probability fraction represents. For example:
6. If there are five green toy cars in a bag, and the probability of choosing a green toy car is , then how many total toy cars are in the bag?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25
All you need to do is set up the proper proportional equation. If 1 of 5 toy cars is green and there is a total of 5 green toy cars, then:
Now we cross multiply the equation to come up with x = 25, choice E.
Probability an Event Won’t Occur
Certain SAT items will ask about the probability of an event not occurring. No sweat. Figure out the probability of the event occurring, and subtract that number from 1:
Probability an event will not occur = 1 – probability an event will occur.
If there is a chance of rain, then the chance of no rain is , or .
Multiple or Unrelated Probabilities
More difficult probability items deal with multiple related and unrelated events. For these items, the probability of both events occurring is the product of the outcomes of each event:
A good example of two unrelated events would be: (1) getting heads on a coin toss and (2) rolling a 5 with a six-sided die. Just find the probability of each individual event, and multiply them together. There’s a 1 in 2 chance of getting heads on a coin and a 1 in 6 chance of rolling a 5. Combining the two gets you:
The same principle can be applied to find the probability of a series of events. Let’s keep it simple and stick to toy cars:
. 14Don has a bag of toy cars divided into 8 blue, 9 green, 4 yellow, and 14 red. Sue bets Don a dollar that she can draw 3 green toy cars in a row. What is the probability that Sue will win the bet?
(A)
(B)
(C)
(D)
(E)
To find the probability of Sue drawing 3 green cars in a row, we need to find the probability of each individual event. The probability of her drawing a green car on her first try is , because there are 9 green cars and 35 total cars (8 blue + 9 green + 4 yellow + 14 red = 35 total).
The probability of her drawing a green car on the second try is slightly different. With one car already removed from the bag, there are only 34 left, and assuming her first try was successful, there are only 8 green cars left. So the probability of drawing a green car the second time is . Follow the same procedure for the probability of choosing a green car on the third try, and we come up with . So the odds of Sue drawing three green cars in a row is:
That’s choice C.
The important point to remember here is that when solving for the probability of a series of events, always assume that each prior event was successful, just as we did in the example above.
Geometric Probability
Another difficult concept the SAT might present is geometric probability. The same basic concept behind probability still applies, but instead of dealing with total outcomes and particular outcomes, you will be dealing with total area and particular area of a geometric figure. There’s nothing fancy here. Just remember this formula and you will be fine:
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