Lab+5+-+Probability

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__Lab 5 - Probability__

__**Exercise 1: Bernoulli Trials**__

A **Bernoulli** **Trial** is an experiment whose outcome is random and can be recorded as either one of two outcomes, typically defined as success or failure. Cite

Heads: 51 Tails: 49
 * RESULTS**:

Frequency of Runs in Coin Flips:
 * Heads ||  || Tails ||   ||
 * 2x || 10 || 2x || 7 ||
 * 3x || 4 || 3x || 2 ||
 * 4x || 0 || 4x || 2 ||
 * 5x || 0 || 5x || 0 ||
 * 6x || 0 || 6x || 0 ||
 * 7x || 0 || 7x || 0 ||
 * 8x || 0 || 8x || 0 ||

The probability of getting a heads is 1/2 (0.5), and tails is 1/2 (0.5). Although not far off, and VERY limitedly, my coin could potentially be pinned as biased EVER so slightly toward heads, but given the closeness of the outcome, it's pretty darn close to being unbiased. The other indication of bias is in the Frequency of Runs; Heads had more of 2-in-a-row runs, however, tails had more higher-frequency runs. In all, my coin seems relatively unbiased.

__**Exercise 2: Probability with Multiple Outcomes**__

- "An experiment is a situation involving chance or probability that leads to results called outcomes." - "An event is one or more outcomes of an experiment." - "An outcome is the result of a single trial of an experiment." - "Probability is the measure of how likely an event is."
 * DEFINITIONS** (Cite):


 * The possible events when rolling a dice include rolling a 1, 2, 3, 4, 5, or 6. **


 * The possible outcomes when rolling a dice include landing a 1, 2, 3, 4, 5, or 6. **


 * The probability of each event occurring is 1/6, or 16.66% **

There isn't //extremely// compelling evidence that the dice was loaded, but if so, I would say loaded against the number 4, since it only had 9 events, while the rest of the events had around 17-22. It could have just been happenstance that the number 4 showed up so significantly fewer times than the other numbers, but perhaps this dice is loaded against the number 4!
 * Results of Dice Rolling Experiment:**
 * Outcome || Events || Total # of Events ||
 * 1 || 17 || 100 ||
 * 2 || 17 || 100 ||
 * 3 || 19 || 100 ||
 * 4 || 9 || 100 ||
 * 5 || 16 || 100 ||
 * 6 || 22 || 100 ||

__**Exercise 3: Probability with Sums of Multiple Outcomes**__

Both Heads: 1/4 = 25% Both Tails: 1/4 = 25% Head & Tail: 2/4, or 1/2 = 50%
 * Probabilities:**

Due to there now being four potential outcomes with two coins, the probabilities shift a bit, and instead of being 50:50 chances of rolling a head or a tail, we now have to combine two independent events. Thus, our probabilities change to the ones listed above, because we multiply the probabilities of the independent outcomes (i.e. for both heads, both tails, we multiply 1/2 by 1/2 to get 1/4).


 * Results of Flipping Two Coins Experiment**
 * Two Heads || 20 || 20% ||
 * Two Tails || 28 || 28% ||
 * One Head & One Tail || 52 || 52% ||

The results came out almost exactly as predicted. I calculated the percentage by doing the frequency / total tosses, so (frequency / 100) for each one.



__**Exercise 4: Probability with Sums of Multiple Outcomes**__

The probability with one die is 1/6, but when adding another die, the probability becomes 1/36:
 * |||| 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: Probablility ||
 * = 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 ||


 * ~ Dice: Frequency ||
 * = 2 ||= 3 ||= 4 ||= 5 ||= 6 ||= 7 ||= 8 ||= 9 ||= 10 ||= 11 ||= 12 ||
 * = 4 ||= 8 ||= 10 ||= 13 ||= 11 ||= 14 ||= 11 ||= 7 ||= 9 ||= 9 ||= 4 ||