Lab+5+Probability


 * Exercise 1: Bernoulli Trials**

A Bernoulli Trial is a random experiment in which two outcomes (success and failure) are possible (http://en.wikipedia.org/wiki/Bernoulli_trial). I performed the Bernoulli trial using a penny and flipped it 100 times. The probability is 50% of achieving heads and 50% of achieving tails, however, in my experience I flipped 45 tails and 55 heads, so heads had the bias in my data by getting 55% of the flips as opposed to 45% for tails.


 * HEADS || TAILS ||
 * 2X 9 || 2X 7 ||
 * 3X 3 || 3X 3 ||
 * 4X 3 || 4X 1 ||
 * 5X || 5X ||
 * 6X || 6X ||
 * 7X || 7X ||
 * 8X || 8X ||

Runs Run y- value is frequency Red = Heads Blue = Tails


 * Exercise 2: Probability with Multiple Outcomes (with Dice)**

An experiment is a situation involving probability. An outcome is the product of one attempt of an experiment. An event is one or more results of an experiment. Probability measures an event's likelihood (http://www.mathgoodies.com/lessons/vol6/intro_probability.html). There are an unlimited possible amount of events, depending how many outcomes are being observed. Possible outcomes are landing on 1, 2, 3, 4, 5, or 6. The probability that each event will occur is 16.7%. No, there is not evidence that the dice is loaded; all of the sides of the dice returned approximately 17 with a derivation of only 2.


 * Outcome || Events || Total # Events ||
 * 1 || 18 || 100 ||
 * 2 || 17 || 100 ||
 * 3 || 17 || 100 ||
 * 4 || 15 || 100 ||
 * 5 || 16 || 100 ||
 * 6 || 15 || 100 ||


 * Exercise 3: Probability with Sums of Multiple Outcomes (Pennies)**

Frequency of 2 Heads, 2 Tails, and 1 Head 1 Tail 2 Heads 2 Tails Heads/Tails

y axis = frequency

The outcome of 2 heads was 32% of the time. The outcome of 2 tails was 32% of the time. The outcome was 1 head and 1 tail 36%. You would calculate the probability of 2 heads as .5 * .5 and of 2 tails is .5 * 5, so it is 25%. The probability of 1 heads and 1 tails is 50% because the first one has an hundred percent chance of being one of the two. The second one has a 50 percent chance of being the other, so 1 * .5 = .5 = 50%.
 * 2 Heads || 32 ||
 * 2 Tails || 32 ||
 * 1 Head 1 Tail || 36 ||


 * Exercise 4: Probability with Sums of Multiple Outcomes (Dice)**


 * The probability of 1 dice for each outcome is 16.7% or 1/6. The probability for each outcome is 1/36 or 2.8%. **


 * <span style="font-family: Arial,Helvetica;">1 (1/6) || <span style="font-family: Arial,Helvetica;">2 (1/6) || <span style="font-family: Arial,Helvetica;">3 (1/6) || <span style="font-family: Arial,Helvetica;">4 (1/6) || <span style="font-family: Arial,Helvetica;">5 (1/6) || <span style="font-family: Arial,Helvetica;">6 (1/6) ||
 * <span style="font-family: Arial,Helvetica;">1 (1/6) || <span style="font-family: Arial,Helvetica;"> 2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% ||  ||
 * <span style="font-family: Arial,Helvetica;">2 (1/6) || <span style="font-family: Arial,Helvetica;"> 2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% ||  ||
 * <span style="font-family: Arial,Helvetica;">3 (1/6) || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% ||  ||
 * <span style="font-family: Arial,Helvetica;">4 (1/6) || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% ||  ||
 * <span style="font-family: Arial,Helvetica;">5 (1/6) || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% ||  ||
 * <span style="font-family: Arial,Helvetica;">6 (1/6) || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% || <span style="font-family: Arial,Helvetica;">2.77% ||  ||


 * Probability** of getting the following sums of individual outcomes:


 * 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 ||
 * 2.77% || 5.55% || 8.33% || 11.11% || 13.88% || 16.66% || 13.88% || 11.11% || 8.33% || 5.55% || 2.77% ||

Measured **Frequency** of getting the following:


 * 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 ||
 * 2% || 7% || 8% || 8% || 12% || 17% || 13% || 10% || 9% || 8% || 6% ||

Sum of 2 Dice y Value = amount of times
 * Frequency (Red) vs. Probability (Blue)**