Week 2, Day 1: Probability
Chapter 14: Math Foundations – Counting Methods & Probability Review
Counting Methods: Probability
The formulas from our memorization sheet are simplified to
facilitate space; the ones in the notes here are a bit more detailed.
For example, P(A) = frequency/outcomes; or in other words,
the probability of any event A is frequency of the desired result over all
possible outcomes. This is the Classical Probability Formula:
(Kaplan has this as number of desired outcomes over number of possible outcomes, but the meaning is the same)
Or P(A) = f ¸ n
1 is the value of absolute certainty, 0 is the value of absolute nonoccurrence. Thus, virtually all probabilities are fractions (or decimals). For example, a fifty percent chance of rain is a ½ probability, a 0.5 probability, or a 50% probability, i.e., fifty chances in one hundred. The higher the probability, the more likely the event will occur; and the lower the probability, the less likely the event will occur.
Contrast with betting odds:
The “odds” are usually the frequency of A versus the
frequency of the other possible events (u = unfavorable events), or
n – f = u odds(A) = f:u
Examples:
The odds of landing on heads in a coin flip: 1:1
The odds of drawing a queen of hearts w/o jokers in the
deck: 1:51
The odds of rolling a three: 1:5
We make two basic assumptions in Classical Probability
(which hold for the GRE):
1. All possible outcomes are taken into account.
In other
words, we do not count unlikely possibilities like a coin landing on edge.
2. Principle of indifference: All possible outcomes are
equally probable.
In other
words, we do not think anyone is cheating.
In reality, these assumptions are not true – people can
cheat, and unlikely events are still possible. However, on the GRE and other
standardized tests, we do not run into these scenarios.
Some of the most basic probabilities are for determined
scenarios involving coin flips, dice rolls, and card pulls from a poker deck
(usually without jokers).
For coin flips, you have two possible outcomes – heads or tails. Thus, for either desired/favored outcome, you have a ½ probability. E.g.: P(heads) = ½
For dice rolls, each number appears once on a six-sided cube-shaped die, i.e. 1, 2, 3, 4, 5, 6.
Thus, for any given number, you have a 1/6 probability.
E.g.: P(5) = 1/6
For even numbers, you have a 3/6 or a ½ probability, and the
same applies to odd numbers. E.g.: P(even) = 3/6 = ½
For other situations, such as numbers greater than 2, you
have a 4/6 or a 2/3 probability, etc. E.g.: P(x>2) = 4/6 = 2/3
For card pulls from a poker deck (minus jokers), you have several choices out of 52 possible cards. There are two colors: red and black. There are four suits: hearts, diamonds, clubs, and spades (♥♦♣♠). Note that hearts and diamonds are red, and clubs and spades are black.
The probability of a red card is half of the deck, or P(red) = 26/52 or ½.
The same is true for black. P(black) = 26/52 =
½
For any given suit, there are 13 cards: 2, 3, 4, 5, 6, 7, 8,
9, 10, J (Jack), Q (Queen), K (King), A (Ace). Thus, the probability of any
suit is 13/52 or ¼. E.g.: P(♥) = 13/52 = ¼
For each face value, there are four such cards; a Queen of
hearts, a Queen of diamonds, a Queen of clubs, and a Queen of spades, so the
probability of a Queen is 4/52 or 1/13. E.g.: P(Q) = 4/52 = 1/13
For each color and face value, there are two cards, i.e. two
red Queens
and two black Queens (or 2s or Aces, etc.). E.g.: P(red Q) = 2/52 =
1/26
But for any one single specific face value and suit, there
is only a 1/52 probability of drawing that card. E.g.: P(Q♥) = 1/52
Other situations outside of classical probability may require you to calculate both the number of desired occurrences and the number of total possible outcomes. Several questions in our book deal with multiple spinners or jars of marbles or other scenarios where we have to do this calculation.
The other formulas are for situations where you want the
probability of more than one event. AND à multiply OR à add NOT à subtract
The first is for one event AND another event where the two events do not change each other; this is known as independent conjunction. P(A & B) = P(A) * P(B)
The second is for one event AND another event where the first event changes the second event; this is known as dependent conjunction. P(A & B) = P(A) * P(B given A)
The third is for one event OR another event where the two events cannot happen simultaneously; this is known as exclusive disjunction. P(A or B) = P(A) + P(B)
The fourth is for one event OR another event where it is possible, but not desired, for the two events to happen simultaneously; this is known as inclusive disjunction. We have to subtract the possibility of both events happening simultaneously to avoid double-counting; this is the inclusion-exclusion principle from set theory. P(A or B) = P(A) + P(B) – P(A & B) or in longer form P(A or B) = P(A) + P(B) – [P(A) * P(B)]
Some events are even more complex, and require both conjunction and disjunction formulas.
The fifth is for when you want to know the probability of an event NOT happening; this is known as negation. Because 1 is absolute certainty, we subtract the possibility of the event happening from 1 to get the probability of the event not happening. P (not A) = 1 – P(A)
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