Basics Of Probability

Probability describes the likelihood of an event occurring. Where:

• Event: Outcome of something
• A: The event under evaluation, is also defined as preferred outcomes.
• P(A): Probability of event A, also defined as $P(A)=\frac{\text{preferred outcomes}}{outcomes}$
• If A and B are independent: P(A and B) = P(A) * P(B)
• If A and B are mutually exclusive: P(A or B) = P(A) + P(B)

Examples

• Coin Flip: For the experiment of flipping two distinguishable coins, with each individual flip resulting in head (H) or a tail (T), we can view the sample space as consisting of the set of all possible 2-string over {H, T}: S = {HH, HT, TH, TT}

• Dice roll

  • A could be rolling a 4: P(A) = 1/6.
  • If A is to roll 4 or 3, we get P(A or B) = 2/6, and so on.
  • If we want to roll 3 and then 6, since the two events are independent, we have that P(A) * P(B) = 1/36

Notes

• Thinking in terms of preferred vs number of outcomes is why combinatorics is very important.
• The sample space S is called the certain event
• (∅) is called the null event.

More concepts:

• Expected Value: E(A) is the outcome we expect when we run the experiment many times.
• PFD: Collection of the probabilities for each possible outcome.
• Events and their complements: The complement of A is everything A is not.

When to multiply or sum probabilities

When dealing with probabilities, you multiply or sum depends on whether the events in question are independent or mutually exclusive.

Multiplication - independent events

You multiply probabilities when you are dealing with independent events (where the occurrence of one does not affect the occurrence of another). For instance, if you are looking for the probability that both of two independent events happen, you multiply their individual probabilities.

For example:

• The probability of flipping a coin and getting heads is $\frac{1}{2}$
• The probability of flipping another coin and getting heads agian is also $\frac{1}{2}$
• The probability of both flips resulting in heads is $\frac{1}{2}$ * $\frac{1}{2}$ = $\frac{1}{4}$

Addition of Probabilities - mutually exclusive events

You sum probabilities when you are dealing with mutually exclusive events, events that cannot happen simultaneously. If you want to find the probability of either of two mutually exclusive events happening, you add their probabilities.

For example:

• The probability of rolling a 1 on a six-sided die is $\frac{1}{6}$
• The probability of rolling a 2 is also $\frac{1}{6}$
• The probability of rolling either a 1 or a 2 is $\frac{1}{6}$ + $\frac{1}{6}$ = $\frac{1}{3}$