probability day 1 (Math for the People)

Main Ideas

Probability measures the likelihood that an event will occur. The notation (P(E)) means “the probability of event (E).”

There are three ways to think about probability:

• Experimental probability: Estimate probability by repeating an experiment and observing its results. The instructor assigns a coin-flip activity: flip a coin 100 times, record the number of heads, and email the result to Dr. Ford with “coin flip experiment” in the subject line.
• Subjective probability: Make an informed estimate when a situation cannot be repeated under identical conditions. For example, predict whether the Kansas City Royals will win a particular game.
• Theoretical probability: When all outcomes are equally likely, calculate probability using: [ P(E)=\frac{\text{number of outcomes in }E}{\text{number of possible outcomes}}. ]

Theoretical Probability and Basic Examples

A fair six-sided die has six equally likely outcomes:

• The probability of rolling a 1 is (1/6).
• The probability of rolling a 3 or 4 is (2/6=1/3).

For a bag containing 16 blue marbles and 4 yellow marbles, the probability of drawing a yellow marble is (4/20=1/5). Probabilities can be written as fractions, decimals, or percentages. An unsimplified fraction also shows the counts used to calculate the probability.

An event is certain if its probability is 1. Drawing a blue or yellow marble from the bag is certain: (20/20=1).

An event is impossible if its probability is 0. Drawing a green marble from the bag is impossible: (0/20=0).

Complements

The complement of an event is the event that it does not happen. If the event is (E), its complement is written (\overline{E}).

The complement rule is:

[ P(\overline{E})=1-P(E). ]

A standard 52-card deck contains four aces. The probability of drawing an ace is (4/52=1/13), or about (7.7\%). The probability of not drawing an ace is (48/52=12/13).

Independent and Dependent Events

Two events are independent when the probability of one is unchanged by whether the other occurs. Examples include two flips of a fair coin, rolling two dice, and the instructor’s example of sunny weather and mail delivery.

For independent events:

[ P(A\text{ and }B)=P(A)\times P(B). ]

For example:

• The probability of getting heads on both of two fair coin flips is (1/2\times1/2=1/4).
• The probability of rolling “snake eyes” (a 1 on each of two fair dice) is (1/6\times1/6=1/36).

Dependent events occur when one event changes the probability of another. Drawing marbles without replacement is an example.

Suppose a bag contains 3 blue and 2 yellow marbles, and two marbles are drawn in order. There are 20 equally likely ordered outcomes, and 6 have blue marbles on both draws. Therefore, the probability of drawing blue twice is (6/20=30\%).

The draws are dependent: after a blue marble is drawn first, 2 of the 4 remaining marbles are blue. Multiplying the probabilities as if they were unchanged would give (3/5\times3/5=36\%), which is incorrect.

“Or” Events and Mutually Exclusive Events

For “(A) or (B),” including cases where both events happen, use:

[ P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B). ]

Subtract the overlap because it is counted in both (P(A)) and (P(B)).

In the two-marble example, the probability that the first marble is blue or the second marble is blue is:

[ 12/20+12/20-6/20=18/20=90\%. ]

Events are mutually exclusive if they cannot happen together, so (P(A\text{ and }B)=0). For mutually exclusive events, the “or” formula reduces to:

[ P(A\text{ or }B)=P(A)+P(B). ]

For example, a single coin toss cannot result in both heads and tails.

Conditional Probability

Conditional probability is the probability of one event given that another is known to have occurred. It is written (P(Y\mid X)), read “the probability of (Y) given (X).”

The formula is:

[ P(Y\mid X)=\frac{P(X\text{ and }Y)}{P(X)}. ]

In the marble example, given that the first marble was blue, the probability that the second marble is blue is (2/4=1/2). Equivalently:

[ \frac{P(\text{both blue})}{P(\text{first blue})} =\frac{6/20}{12/20} =\frac{1}{2}. ]

The instructor also applies conditional probability to a home pregnancy-test table:

• The probability someone is not pregnant given a positive test is (5/75\approx6.7\%).
• The probability of a positive test given the person is not pregnant is (5/19\approx26\%).

These are different conditional probabilities because each condition identifies a different group.

Speaker and Source

Dr. Ford is the instructor and narrator. No other speakers or sources are featured.