# Probability theory

Probability theory – the branch of mathematics concerned with analysis of random phenomena. As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data.

Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are

• the law of large numbers
• the central limit theorem

Statistics – statistics offers mathematical tools that can account for possible confounders. That allows scientists to see how much a change in one variable might be linked to differences in something else.

That’s engineering

• data – Facts and statistics collected together for analysis but not necessarily organized in a way that give them meaning. For digital information (the type stored by computers), those data typically are numbers stored in a binary code, portrayed as strings of zeros and ones.
• statistical significance – In research, a result that appears reliable — or significant — from a mathematical point of view. Findings that are statistically significant have a reduced likelihood that the apparent link between two or more conditions is a fluke, or merely due to chance.

Engineering ideas

• random, chance, analysis, statistics,quantitative, data, sets, event, sequence, pattern, predict, representative, limit, theorem

Do It
Challenges for you to work on…

• penny toss – heads or tails. Toss a penny 10 times and record the number of heads and tails. What percent were heads? Toss the penny 10 more times and record the results. Do this until you have 100 tosses total. Compare the percentage heads for each group of 10 tosses. Compare this with the percentage heads for the entire 100. What can you conclude from this information? What is the probability that the next toss will be heads?
• roll of the die – similar to the penny toss but now the result can be a number from 1 to 6 (number of faces on the die, each with a number).