Probability: The study of Randomness Chapter 5: Probability: The study of Randomness

With probability, if I do something long enough I should get similar results. Flip a coin, frequency of getting heads is erratic if you only flip 2,5, or 10 times. Sometimes probability = .2, .6, … Flip 10,000 times, you should get .5

“Random” does not mean “haphazard “Random” does not mean “haphazard.” Random is a description of a kind of order that emerges only in the long run. In daily life, random things occur to us but we rarely see enough repetitions of the same random phenomenon to observe long term regularity.

Ex) On your way to school you pass a man in a clown suit at a bus stop Ex) On your way to school you pass a man in a clown suit at a bus stop. You don’t see him again until next week. Then the week after you realize you only see him on Tuesday because clown school is on Tuesdays.

We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

Probability must have independence Probability must have independence. That is, the outcome of one trial must not influence the outcome of another.

The Sample Space S of a random event is the set of all possible outcomes. Ex) Toss a coin; S = {heads, tails} or S= {H,T}

Ex) Flip a coin + Roll a die. S= {H1,H2,H3,…. T1,T2,…T6} Tree Diagram: 4 H4 5 H5 6 H6 T1 1 T2 2 T T3 3 4 T4 5 T5 6 T6

Multiplication Principle If you can do one task in ‘a’ number of ways and a second task in ‘b’ numbers of ways, then both tasks can be done in a x b number of ways.

Replacement / Without Replacement Ex) You have 10 numbers 0-9 in a hat. You choose three with replacement: - 10 x 10 x 10 = 1000 3-digit numbers Without replacement: - 10 x 9 x 8 = 720 3-digit numbers