Lab+5+Probability

=__**Exercise 1: Bernoulli Trials**__= -The Bernoulli trail is an experiment done to determine the probability or statistics or a certain thing. The results of this experiment are random and can result in either "failure" or "success". Success and failure are determined from the response of a yes or no question. For example, did the coin land on tails? **[]** __**Table of the Frequency of Runs in Coin Flips**__ __**Graph**__ -The probability of a coin landing on heads or tails should be 50% each due to the fact that there are only two different outcomes possible. However, from my graph there is a slight bias towards tails because the results came out to be a probability of 54% for tails and 46% for heads over the 100 attempts.
 * **HEADS** || **TAILS** ||
 * **2X: 10** || **2X: 6** ||
 * **3X: 1** || **3X: 2** ||
 * **4X: 2** || **4X: 2** ||
 * **5X: 0** || **5X: 1** ||
 * **6X: 0** || **6X: 0** ||
 * **7X: 0** || **7X: 1** ||
 * **8X: 0** || **8X: 0** ||

=__**Exercise 2: Probability with Multiple Outcomes (Dice)**__= [] -When rolling a dice there is one possible event because there can only be one outcome. For example, when rolling a dice and landing on 2, 2 is the single event that results. -When rolling a dice the possible outcomes are 6. This is because there are 6 different numbers that the dice can land on. -In 100 throws of the dice there is a probability of 16.6% that each number or outcome will occur. This is determined by the fact that there are 6 different outcomes and only one roll at a time so 1/6=.16. __**Table of Probability with Dice**__ -The following table shows the probability of each number occuring when rolling the dice 100 times. The information from the table might lead one to say that the dice was slightly loaded because the numbers 3 and 6 showed up above the average while 1 and 2 showed up below the average of 16.6%. =**__Exercise 3: Probability with Sums of Multiple Outcomes (Pennies)__**= __**Graph of Results of Pennies**__ -Based on the results, the outcome of 2 heads resulted 25% of the time. This was found by 25/100= .25. -Based on the results, the outcome of 2 tails resulted 27% of the time. This was found by 27/100= .27. -Based on the results, the outcome of 1 head and 1 tail is 48% of the time. This was found by 48/100=.48. -In order to calculate the probability for flipping two coins one must recognize the possible outcomes. The total possible outcomes are HH, HT, TH, TT. Of these four possible outcomes there are 2 out of 4 that will have a head and a tail giving a probability of 50%. However, there is only a 1 out of 4 chance that will result in two heads or two tails so the probability of each happening is 25%. =__﻿Exercise 4: Probability with Sums of Multiple Outcomes (Dice)__= -The probability of each outcome for 1 dice is 16.6%. This is was found by dividing the 1 event by the 6 possible outcomes 1/6=.166. -The probability of each outcome for 2 dice is 2.7%. This is found by multiplying the probability by the probability because now there is 2 dice instead of one, 1/6x1/6=.027. __**Table of Probability of Sums**__
 * 1. Experiment:** An experiment involves a situation in which there is a certain probability or chance that results in an outcome. An example involving dice would be rolling dice 100 times to see the probability of each number appearing.
 * 2. Outcome:** An outcome is simply the result after a trial has been completed in an experiment. An example involving dice is rolling a dice and landing on either 1,2,3,4,5, or 6. All of these numbers are therefore outcomes.
 * 3. Event:** An event is one or more outcomes that result from a certain experiment. An example involving dice would be rolling a dice and landing on 2. Landing on 2 is therefore one event.
 * 4. Probability:** Probability is defined as the measure or chance of how likely an event is to occur. An example involving dice would be that the probability of landing on the number three is 1/6 since there are 6 different numbers on a dice.
 * **Outcome** || **Events** || **Total # Events** ||
 * **1** || **14** || **14/100= 14%** ||
 * **2** || **14** || **14/100= 14%** ||
 * **3** || **20** || **20/100= 20%** ||
 * **4** || **16** || **16/100= 16%** ||
 * **5** || **16** || **16/100= 16%** ||
 * **6** || **20** || **20/100= 20%** ||
 * __Table of Results of Pennies__**
 * **2 Heads** || **25 times** ||
 * **1 Head + 1 Tail** || **48 times** ||
 * **2 Tails** || **27 times** ||
 * || **1 (1/6)** || **2 (1/6)** || **3 (1/6)** || **4 (1/6)** || **5 (1/6)** || **6 (1/6)** ||
 * **1 (1/6)** || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 ||
 * **2 (1/6)** || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 ||
 * **3 (1/6)** || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 ||
 * **4 (1/6)** || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 ||
 * **5 (1/6)** || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 ||
 * **6 (1/6)** || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 || 1/36 ||

__**Dice Probability (Sums of Individual Outcomes)**__
 * **2** || **3** || **4** || **5** || **6** || **7** || **8** || **9** || **10** || **11** || **12** ||
 * 1/36 || 2/36 || 3/36 || 4/36 || 5/36 || 6/36 || 5/36 || 4/36 || 3/36 || 2/36 || 1/36 ||

__**Probability vs. Frequency**__ -The blue line shows the probability and the red line shows the frequency.
 * __Measured Frequency__**
 * **2** || **3** || **4** || **5** || **6** || **7** || **8** || **9** || **10** || **11** || **12** ||
 * 5/100 || 7/100 || 7/100 || 11/100 || 9/100 || 16/100 || 17/100 || 8/100 || 16/100 || 3/100 || 1/100 ||