Stat 11 Week 6 & 8
Random Variables and Probability Rules - Part 1
Prof. Suzy Thornton
1. Random Phenomena
Random variables result in random events. Each random variable has a sample space of outcome events. A trials results in an observed event.
Example: Rolling a six-sided die
The random event is the face of the die that lands upward.
The sample space of outcomes is this six faces of the die.
A trial represents the act of physically rolling a six-sided die and producing an observation of this random phenomena.
Example: Flipping an unfair coin
What is the random event?
What is the sample space of outcomes?
A trial represents the act of physically flipping the coin and producing an observation of this random phenomena.
2. Law of Large Numbers
As we repeat a random process over and over again, the proportion of times that an event occurs eventually settles down to a single probability that describes the relative frequency with which this event occurs. In short, the Law of Large Numbers (LLN) asserts that relative frequencies settle down to a single value in the long run.
Empirical probability - uses the laws of probability to describe the long-run behavior of repeated observations of a random event.
Example: Rolling a six-sided die
# Consider a six sided die
S = c(1,2,3,4,5,6)
# Roll the die once
sample(S, size=1, replace=TRUE)## [1] 6Q) What’s the probability that the die lands on a “six”?
- \(1/6=\) 0.1666667
Q) What’s the probability that we roll the dice again and it lands on a “six” again?<>
Note: There is no such thing as a “law of averages”. This is also known as “the gambler’s fallacy”.
# Roll the die 10000 times
trials = 10000
observations = sample(S, size=trials, replace=TRUE)
observations## [1] 1 4 1 2 4 3 3 5 3 6 5 4 4 4 1 1 6 2 2 3 4 3 1 4 5 3 4 3 1 2 1 1 2 3 2 4
## [37] 2 1 2 4 2 5 4 5 6 3 2 1 6 2 1 3 4 3 5 3 4 4 2 3 2 1 5 2 1 5 3 3 6 6 5 6
## [73] 2 6 1 4 1 1 4 4 5 2 2 3 6 6 3 3 2 4 2 1 4 6 6 4 2 2 6 4 2 6 2 2 5 6 5 2
## [109] 4 1 1 5 2 2 1 3 2 6 2 2 4 3 2 5 5 2 4 4 4 1 6 6 5 5 4 1 6 6 4 2 6 2 2 3
## [145] 5 2 4 2 1 4 3 6 2 6 2 1 5 6 5 5 6 3 5 3 6 4 5 4 2 2 3 5 3 6 1 3 3 5 4 1
## [181] 5 2 6 2 4 5 4 1 1 5 4 1 4 1 1 4 6 5 5 5 1 4 5 4 3 5 3 5 2 1 6 4 6 2 3 3
## [217] 6 1 6 3 5 1 5 5 5 4 3 2 5 3 5 4 6 2 1 1 3 6 6 3 6 4 1 3 3 2 2 5 3 5 4 5
## [253] 6 1 3 4 3 5 2 6 2 6 3 1 5 6 2 6 3 6 4 6 3 1 1 5 3 1 1 4 5 1 6 4 4 2 4 2
## [289] 2 6 1 4 2 2 4 5 5 1 5 1 4 3 1 3 5 5 3 4 6 1 2 3 2 4 1 2 6 6 3 3 5 5 1 4
## [325] 3 6 5 2 2 4 2 5 6 3 5 6 6 3 2 5 3 3 3 1 1 6 6 6 4 1 1 5 5 1 3 4 1 4 4 3
## [361] 2 1 4 2 6 6 2 6 5 3 1 2 2 3 2 3 6 5 1 1 6 5 3 2 6 1 3 6 3 4 4 1 4 2 4 5
## [397] 3 2 3 5 5 1 4 1 4 5 4 5 3 5 4 1 1 2 4 2 3 3 2 6 6 5 6 1 2 5 5 5 1 4 2 4
## [433] 2 5 3 6 3 5 1 2 1 1 2 2 4 2 5 2 1 1 6 5 2 4 1 6 6 4 1 6 2 5 3 1 5 5 2 5
## [469] 3 2 4 3 3 6 6 1 5 5 1 1 3 1 3 5 4 5 3 5 1 2 5 2 2 3 4 2 4 5 1 2 5 1 3 2
## [505] 4 3 4 4 6 6 6 4 2 2 3 6 1 6 5 1 2 3 2 6 3 2 2 1 2 3 4 5 4 5 1 4 1 6 3 2
## [541] 2 3 2 3 3 5 2 3 3 5 4 5 6 5 2 4 3 1 3 3 5 5 5 5 2 1 3 3 2 3 5 3 5 3 1 1
## [577] 2 4 1 4 1 1 5 4 1 3 2 2 1 1 5 3 3 4 2 2 5 1 4 1 1 6 2 4 6 1 2 5 6 3 6 2
## [613] 5 1 3 2 3 3 6 6 2 4 6 1 2 3 1 3 6 1 2 1 3 2 6 2 2 3 1 1 2 6 4 3 3 6 3 2
## [649] 3 5 2 4 6 2 2 5 6 5 4 2 6 4 4 3 2 2 2 3 5 5 3 1 2 6 5 3 3 3 1 4 3 6 2 5
## [685] 5 6 6 6 1 5 5 3 1 4 1 1 6 2 4 3 5 5 2 3 2 5 6 1 3 6 5 1 4 5 5 6 1 6 3 5
## [721] 4 5 1 2 2 1 1 4 1 1 6 2 5 5 6 1 2 1 3 3 5 2 2 5 3 4 3 1 1 1 2 4 1 6 1 5
## [757] 6 1 3 4 3 1 6 4 6 1 1 2 3 6 1 5 5 1 6 6 2 4 1 2 2 1 6 5 4 6 5 6 4 4 1 5
## [793] 4 1 6 4 5 6 3 6 1 4 3 6 5 6 5 5 3 5 6 6 5 2 6 3 5 3 1 6 6 5 3 3 6 6 6 3
## [829] 4 2 4 3 5 6 2 3 5 4 6 6 4 3 6 2 2 3 1 4 3 3 3 2 3 4 5 5 4 1 4 3 1 6 2 6
## [865] 4 4 2 5 6 6 2 6 2 3 2 6 3 6 1 1 1 2 1 4 4 4 4 4 4 1 3 4 3 3 6 5 6 3 3 1
## [901] 6 3 5 1 4 4 3 3 4 1 5 6 5 5 6 2 1 1 1 2 3 1 1 3 1 1 3 2 3 3 1 5 2 1 5 6
## [937] 5 4 2 4 4 1 2 4 1 3 1 3 2 2 3 1 1 5 3 1 5 2 2 1 4 3 4 5 5 2 4 4 6 3 2 4
## [973] 1 2 3 5 2 1 3 4 3 1 2 4 2 6 5 2 1 2 4 3 5 5 2 4 3 3 2 5 5 2 5 6 1 1 6 2
## [1009] 5 4 5 4 3 6 4 1 2 3 2 4 5 3 5 5 3 2 3 4 3 2 1 6 4 2 5 1 4 2 5 5 1 4 6 2
## [1045] 4 5 5 6 1 4 4 5 4 2 5 2 3 5 4 6 3 6 4 4 3 4 6 4 4 4 4 1 3 1 5 2 6 6 2 4
## [1081] 5 4 2 4 3 2 3 2 6 6 5 6 3 2 5 1 4 5 3 3 1 4 6 1 2 2 1 4 6 2 4 5 6 4 1 6
## [1117] 3 6 1 6 6 6 2 4 3 2 6 2 3 4 5 6 3 3 4 6 4 4 4 2 5 4 2 6 6 4 3 4 3 4 4 4
## [1153] 6 5 5 5 2 6 3 2 4 1 4 2 5 5 3 2 3 5 1 6 6 3 3 1 5 5 3 3 2 6 1 6 1 3 2 2
## [1189] 4 4 6 3 5 2 1 6 2 1 4 6 5 1 5 6 5 6 2 6 5 6 1 2 2 4 4 5 3 2 1 4 2 1 5 6
## [1225] 6 3 3 1 1 4 1 3 1 5 5 6 1 3 5 4 2 6 4 3 5 3 1 2 1 2 5 5 4 4 4 3 6 1 2 5
## [1261] 2 4 5 1 3 2 5 4 2 1 3 3 4 4 5 5 5 2 4 6 3 3 2 2 4 1 1 2 1 5 1 2 3 4 6 2
## [1297] 2 4 1 3 2 3 4 2 1 5 2 6 4 2 4 3 1 4 4 5 5 6 2 1 4 3 1 1 2 6 6 4 4 5 6 5
## [1333] 2 1 5 3 5 6 2 2 2 1 2 2 1 2 2 6 6 1 4 3 2 2 3 5 1 5 2 5 5 2 6 6 1 4 2 1
## [1369] 6 3 5 2 6 5 5 2 1 1 2 1 4 5 3 1 4 6 2 3 4 5 1 6 3 3 1 5 5 6 2 5 3 3 5 4
## [1405] 5 4 4 4 5 2 5 4 3 4 6 6 1 4 6 2 1 5 2 4 6 2 2 5 4 3 4 3 3 2 1 2 5 6 2 4
## [1441] 1 6 1 6 1 6 3 2 6 6 1 3 6 4 2 6 3 5 3 5 4 2 4 6 3 1 3 4 4 2 4 1 5 6 1 3
## [1477] 2 1 1 5 5 5 1 6 1 6 4 3 3 1 5 6 2 1 1 2 3 1 3 3 3 5 5 6 2 5 6 1 6 3 5 4
## [1513] 6 5 3 6 4 4 3 5 6 2 3 3 3 2 4 1 4 2 3 4 6 1 2 6 2 2 1 2 5 1 6 4 4 2 2 4
## [1549] 2 1 6 5 3 4 2 1 2 5 2 4 4 1 3 2 5 6 1 4 6 6 6 4 5 2 5 5 5 6 5 2 5 4 4 1
## [1585] 4 3 6 3 4 4 5 2 6 1 6 2 5 2 4 4 3 1 2 2 6 2 6 2 3 1 4 2 6 6 6 1 6 5 5 6
## [1621] 3 5 5 1 4 5 5 3 2 4 3 6 3 5 1 4 4 2 3 3 3 1 2 1 1 5 4 5 1 4 1 2 3 1 5 5
## [1657] 3 6 2 1 3 2 4 3 1 4 2 6 2 3 6 6 6 6 2 2 2 1 4 1 5 5 4 4 4 1 2 3 3 4 2 2
## [1693] 3 2 3 6 6 4 5 4 6 1 2 2 4 5 3 2 3 4 6 5 2 6 3 6 6 3 5 5 4 5 4 3 2 1 3 3
## [1729] 5 1 5 5 5 1 1 6 1 2 3 4 1 4 2 4 5 6 6 6 3 1 6 1 4 3 6 6 3 3 2 2 4 2 5 5
## [1765] 6 4 3 5 2 6 6 4 2 5 6 5 6 4 1 5 6 2 1 5 4 1 5 1 5 5 1 5 2 6 1 3 5 6 3 3
## [1801] 6 1 3 5 2 3 2 3 5 4 4 5 6 6 1 4 3 4 1 1 2 4 5 5 1 1 6 5 1 4 4 1 2 2 6 3
## [1837] 4 1 4 5 4 3 1 5 3 2 5 3 6 3 5 3 1 1 6 6 1 5 4 3 6 5 4 6 5 1 5 3 2 6 3 5
## [1873] 3 6 4 2 5 4 1 2 3 1 4 5 5 6 2 4 3 3 3 2 2 2 4 4 2 6 6 5 4 2 5 1 2 4 1 6
## [1909] 3 1 2 3 3 3 4 5 1 4 4 2 4 5 1 2 6 1 3 4 3 3 5 2 6 5 4 4 1 2 3 2 4 3 3 6
## [1945] 3 4 4 5 2 3 3 6 1 4 4 2 3 6 5 2 5 1 6 1 1 2 1 6 5 2 4 2 5 3 3 5 6 4 3 4
## [1981] 4 6 4 3 2 6 3 5 4 6 4 2 1 3 2 2 1 2 5 2 4 2 6 4 5 2 6 4 2 5 3 6 5 6 5 2
## [2017] 1 6 4 1 6 2 2 1 2 5 2 2 3 4 4 1 4 4 1 3 1 4 3 5 4 5 5 4 2 2 1 4 6 6 3 4
## [2053] 1 4 1 6 3 1 2 5 3 3 1 2 2 4 4 4 4 2 3 4 2 3 1 4 2 5 5 5 6 5 2 5 1 2 2 6
## [2089] 1 1 5 5 1 4 6 6 6 4 3 6 6 4 1 1 3 5 1 6 4 6 3 2 2 1 3 6 1 1 1 3 1 5 5 1
## [2125] 2 4 4 3 2 2 6 4 6 1 5 4 1 2 2 4 3 1 6 4 1 1 5 2 6 6 5 2 1 3 1 5 3 1 4 1
## [2161] 4 5 5 3 2 6 4 6 2 6 1 4 3 2 5 2 1 3 5 1 4 5 4 5 4 5 3 5 5 2 2 4 5 4 5 3
## [2197] 3 3 2 4 5 5 3 4 3 1 2 2 1 2 3 2 3 6 5 5 1 3 3 5 5 1 2 6 4 3 6 4 3 2 3 6
## [2233] 6 3 3 3 2 4 5 5 5 2 5 6 5 5 2 6 3 4 2 6 2 5 3 1 3 3 4 4 2 1 4 3 6 5 2 4
## [2269] 2 1 4 6 2 6 2 4 4 6 4 1 2 3 3 1 5 5 6 2 4 2 5 3 1 1 5 6 1 4 2 4 3 1 3 4
## [2305] 3 4 1 1 2 3 2 1 4 1 3 6 4 6 6 2 2 6 3 4 1 1 1 6 4 5 3 3 2 4 3 4 3 6 3 6
## [2341] 6 5 2 1 5 4 3 5 2 2 2 2 2 2 6 6 2 2 1 2 4 4 6 4 6 3 2 2 6 5 5 1 4 4 2 2
## [2377] 5 2 3 3 5 4 4 4 2 4 3 2 4 3 2 2 3 2 2 2 1 3 3 4 6 4 6 4 2 1 6 1 4 2 2 2
## [2413] 2 5 2 4 3 5 6 1 1 5 3 6 1 5 5 2 5 6 2 3 3 6 5 4 1 6 3 1 2 6 3 5 4 1 6 4
## [2449] 4 2 2 3 1 4 5 3 3 1 5 3 5 1 6 2 2 1 2 2 4 3 1 3 4 2 1 6 6 4 6 4 1 1 4 5
## [2485] 1 5 1 4 5 6 1 5 3 4 6 4 5 4 1 3 4 5 4 3 5 6 1 6 5 2 5 4 4 5 5 5 3 3 2 6
## [2521] 3 2 5 1 5 6 1 3 6 6 1 4 6 2 5 6 1 5 6 4 6 6 3 5 3 3 4 1 5 5 3 4 4 1 2 3
## [2557] 6 1 2 6 6 5 5 1 4 5 3 6 4 5 4 1 3 5 3 1 2 1 5 1 3 2 2 4 1 6 5 3 4 1 2 6
## [2593] 5 4 6 1 1 2 2 5 6 2 2 3 1 5 3 5 2 6 3 5 4 4 6 3 5 5 2 2 1 5 4 6 3 1 5 1
## [2629] 6 1 5 3 5 6 2 2 3 4 6 1 5 1 3 5 5 6 6 4 6 3 3 1 4 2 1 5 5 5 5 1 4 2 6 5
## [2665] 4 5 3 4 3 6 6 4 6 5 5 5 1 2 5 5 2 1 4 2 3 6 6 6 2 3 5 2 3 4 5 4 1 6 3 6
## [2701] 5 6 4 4 1 1 3 1 5 4 5 1 5 1 4 1 6 2 5 2 5 4 3 1 4 3 1 2 3 6 1 4 2 2 1 5
## [2737] 1 5 1 6 1 5 1 4 4 6 5 4 2 6 1 4 3 3 4 3 4 3 4 6 4 5 1 3 1 3 2 5 5 3 3 3
## [2773] 5 5 2 5 6 2 5 6 5 5 6 4 2 3 4 6 4 4 6 5 5 4 2 4 2 3 5 5 6 5 1 2 6 2 4 3
## [2809] 5 4 6 2 2 2 4 2 2 5 6 1 2 6 1 3 2 6 6 2 4 5 6 2 4 3 4 3 2 5 6 2 4 6 4 3
## [2845] 3 6 1 4 5 4 2 3 6 5 4 5 5 2 4 4 1 1 4 2 6 1 1 3 5 5 4 5 1 5 3 5 5 5 4 5
## [2881] 1 3 1 6 2 3 5 3 4 6 5 6 4 5 6 2 1 4 6 6 1 1 6 1 6 5 3 6 3 1 1 1 1 4 1 5
## [2917] 2 3 6 4 3 5 1 1 5 2 2 1 4 4 6 1 2 4 6 2 4 2 5 3 5 5 1 1 3 1 2 1 6 5 6 3
## [2953] 6 2 5 3 3 3 6 2 5 2 6 4 4 4 3 1 5 4 2 6 3 1 6 6 3 5 1 1 2 5 2 4 5 4 6 3
## [2989] 4 3 3 5 6 5 3 1 5 3 4 6 2 2 5 2 5 6 5 6 5 4 1 3 1 6 1 5 6 5 5 6 5 1 4 3
## [3025] 6 6 5 1 4 6 6 2 1 3 3 4 2 4 5 2 3 6 2 3 3 1 2 6 5 1 6 2 4 2 2 4 4 6 2 5
## [3061] 4 3 4 3 4 2 5 1 6 1 5 2 6 1 2 6 5 1 6 4 5 2 2 1 4 1 6 3 4 5 1 1 3 1 5 2
## [3097] 5 1 6 3 3 6 4 3 4 6 4 4 4 1 2 3 2 5 2 5 1 3 2 4 2 2 6 6 1 1 5 6 4 5 1 5
## [3133] 2 1 6 3 1 2 3 2 3 6 2 2 2 4 4 5 4 2 4 5 3 1 6 2 1 3 2 4 6 2 6 2 2 5 5 2
## [3169] 3 1 5 3 2 5 4 3 2 6 3 2 6 3 2 6 4 2 5 5 1 6 4 4 4 1 1 1 4 2 2 4 4 6 3 2
## [3205] 1 1 6 4 3 3 1 4 4 3 3 1 5 4 3 3 3 5 1 4 6 1 3 2 2 5 1 4 6 3 2 3 1 1 5 2
## [3241] 6 1 6 4 1 1 2 4 4 4 1 1 5 5 5 5 1 5 4 2 5 2 6 3 4 2 2 5 5 5 5 4 3 4 4 5
## [3277] 5 2 4 4 1 1 6 4 1 4 3 1 5 3 4 2 6 1 2 1 4 5 1 2 5 5 2 4 6 1 2 2 2 1 6 5
## [3313] 6 5 3 4 2 1 2 4 3 1 2 1 5 3 1 1 5 1 1 5 1 1 4 3 3 5 6 6 2 6 3 2 3 5 6 5
## [3349] 6 4 6 5 1 4 1 3 2 2 1 6 6 4 4 2 4 6 5 1 6 2 5 6 3 4 4 4 3 3 2 2 5 3 4 2
## [3385] 1 3 2 2 4 6 3 5 5 3 2 4 4 1 3 3 5 2 5 1 1 2 1 5 2 5 1 6 3 2 1 3 5 3 1 3
## [3421] 1 4 6 3 2 4 1 4 6 3 3 3 4 5 2 4 2 2 4 5 3 5 5 3 4 1 1 5 5 1 4 5 5 6 3 5
## [3457] 4 5 2 4 4 1 1 2 3 4 4 2 6 5 4 5 3 2 1 2 4 3 6 3 4 5 2 5 5 1 3 3 1 3 1 3
## [3493] 5 6 5 5 3 6 6 2 1 6 5 3 4 3 6 2 4 3 2 3 1 2 4 6 1 5 1 6 5 5 5 1 4 2 6 4
## [3529] 3 2 3 2 3 1 4 2 4 3 4 4 3 6 2 2 4 4 4 6 4 2 3 6 6 2 4 4 4 5 1 2 6 4 4 6
## [3565] 4 4 2 5 1 2 4 4 3 2 2 3 6 3 3 6 6 3 5 2 1 6 6 3 3 2 5 6 1 6 4 3 2 4 2 3
## [3601] 4 3 3 2 5 3 1 1 6 2 6 3 3 6 5 2 4 6 6 5 5 4 4 5 4 4 1 3 5 1 6 4 6 5 2 6
## [3637] 6 3 2 2 6 4 5 2 1 6 3 3 4 1 2 2 6 1 6 4 4 1 5 6 4 3 4 5 4 5 5 3 1 5 2 2
## [3673] 6 4 4 5 6 6 3 6 1 5 2 3 2 3 4 3 3 2 5 3 4 3 3 5 4 4 3 2 5 5 6 2 2 1 1 6
## [3709] 3 1 2 1 4 1 6 2 2 5 4 2 5 6 3 5 4 4 1 1 4 5 6 3 3 3 2 5 5 2 1 1 1 1 4 5
## [3745] 3 1 2 1 1 5 5 5 6 5 2 5 2 5 2 4 4 2 2 6 4 1 2 4 5 2 1 1 5 1 5 5 2 5 4 6
## [3781] 3 2 5 5 4 2 3 6 5 5 4 3 3 6 5 6 1 4 6 6 2 3 2 2 4 5 2 6 3 5 3 6 3 2 1 3
## [3817] 4 6 1 1 1 6 3 2 5 6 4 6 2 3 4 2 1 5 1 1 1 6 6 6 6 3 5 5 2 3 3 3 1 2 4 2
## [3853] 2 5 5 2 4 5 5 1 4 5 4 5 3 3 6 1 1 6 2 5 6 4 6 4 6 6 5 3 1 3 6 1 5 5 5 6
## [3889] 2 2 4 2 4 1 1 4 4 4 1 3 3 6 2 5 5 3 4 1 4 6 4 2 3 3 4 5 3 1 4 2 6 2 6 1
## [3925] 5 5 1 5 2 4 1 4 3 5 6 1 1 2 4 5 5 6 2 5 1 4 1 1 6 1 6 4 4 4 2 2 4 4 5 6
## [3961] 6 1 4 6 5 5 3 1 1 4 4 6 5 4 3 2 6 4 6 5 3 1 6 5 2 5 4 1 5 1 1 5 3 1 2 6
## [3997] 2 1 4 3 2 6 1 5 5 4 5 4 4 5 5 5 2 2 4 1 4 1 2 4 2 1 6 1 4 2 2 3 2 4 6 5
## [4033] 6 4 1 6 5 5 6 4 1 3 6 2 2 3 2 5 3 1 2 6 4 3 6 3 6 5 2 3 6 5 4 4 1 6 6 4
## [4069] 6 5 3 2 3 3 1 2 6 1 2 3 5 5 2 1 2 5 1 2 6 6 2 2 3 4 5 4 1 2 6 1 3 2 3 5
## [4105] 6 4 6 2 4 1 1 5 6 2 4 4 4 2 5 4 1 2 4 5 4 6 6 1 6 4 3 5 2 2 1 2 5 6 1 4
## [4141] 4 2 6 3 4 3 6 1 6 1 2 1 5 5 3 1 3 3 6 1 1 2 5 6 3 4 5 6 3 4 1 4 3 4 1 5
## [4177] 4 3 1 4 1 5 2 3 4 4 6 2 1 2 6 4 2 6 5 4 3 6 2 4 4 2 4 2 2 4 5 4 1 4 5 2
## [4213] 2 5 3 4 2 4 3 3 1 3 2 5 4 6 6 1 1 4 5 5 5 5 6 6 3 4 6 3 6 5 2 4 6 4 1 1
## [4249] 3 5 3 2 1 5 1 6 1 4 1 4 2 4 2 1 2 1 4 6 6 5 1 4 1 3 1 5 3 6 3 6 1 6 2 6
## [4285] 3 6 3 4 4 3 4 3 4 2 5 5 2 5 2 1 6 2 4 4 6 6 5 4 3 1 3 1 5 4 1 6 4 2 2 1
## [4321] 1 2 5 5 5 5 2 4 2 1 1 3 2 3 6 4 5 3 2 3 6 1 5 2 3 4 5 2 4 6 2 6 2 1 2 4
## [4357] 3 2 4 2 1 5 4 5 2 4 5 2 5 3 1 4 5 4 2 4 6 3 2 4 1 2 1 3 3 2 2 1 5 4 4 1
## [4393] 3 3 2 6 6 4 2 2 6 4 6 4 1 5 6 6 4 4 2 6 3 1 6 3 6 1 2 5 2 1 4 6 3 1 4 2
## [4429] 3 3 5 5 4 1 6 2 3 4 1 5 5 3 4 6 3 6 2 1 5 4 1 4 1 5 4 2 1 6 1 5 4 2 3 1
## [4465] 5 5 4 2 3 1 6 3 4 3 3 1 4 2 2 4 1 6 1 3 1 5 4 6 5 1 5 6 3 3 3 3 6 4 2 1
## [4501] 6 6 6 4 3 5 6 4 1 5 2 4 5 5 5 4 6 3 2 4 3 5 6 3 3 4 3 5 6 5 3 1 1 4 1 5
## [4537] 2 1 3 1 5 6 2 3 1 2 6 2 6 5 2 1 1 5 3 3 2 5 3 6 2 5 4 5 3 3 2 1 2 2 4 1
## [4573] 2 4 3 5 1 3 1 6 6 4 2 6 1 1 2 3 5 4 3 3 1 6 2 2 1 1 1 5 4 2 4 2 4 2 1 2
## [4609] 5 3 4 3 2 2 4 4 3 2 1 1 3 5 4 3 6 5 2 5 5 5 3 4 5 1 6 6 3 2 5 2 2 4 2 1
## [4645] 1 3 5 4 2 6 6 2 4 1 1 5 2 6 5 1 3 5 5 1 6 1 4 1 4 3 3 4 1 3 3 3 4 3 2 3
## [4681] 1 6 2 2 1 2 6 1 1 5 4 4 4 3 6 4 1 3 2 5 5 1 6 3 2 5 5 4 5 2 4 6 1 2 5 6
## [4717] 6 1 5 6 6 6 6 2 5 2 4 5 1 4 4 5 3 1 6 1 6 3 5 6 2 5 2 1 3 5 4 2 4 4 1 3
## [4753] 5 4 1 6 1 5 4 5 5 3 1 6 2 5 6 1 4 5 4 5 6 3 4 3 2 6 1 5 4 6 4 5 5 3 3 4
## [4789] 3 6 4 5 1 4 4 5 6 5 3 3 5 2 5 3 3 4 6 2 3 6 5 2 1 3 5 5 1 1 4 1 1 1 3 5
## [4825] 1 4 2 1 2 2 1 1 3 4 4 4 2 1 6 4 3 5 2 1 6 4 1 1 2 4 6 4 2 3 1 4 3 3 5 3
## [4861] 6 2 5 2 5 6 5 4 2 3 1 1 1 1 5 1 1 2 5 5 6 5 2 6 1 5 5 4 2 5 6 4 5 3 3 3
## [4897] 4 6 1 5 6 4 3 4 4 6 1 1 2 6 1 6 3 1 6 4 4 2 3 6 2 3 6 1 2 4 5 1 3 6 3 1
## [4933] 4 6 1 2 6 5 5 1 1 1 2 4 2 6 6 6 4 1 2 1 4 6 1 4 5 6 5 5 6 4 6 3 5 6 5 4
## [4969] 6 6 5 4 1 1 2 2 5 6 2 1 3 1 3 6 2 4 1 4 6 1 3 4 1 1 4 5 3 4 6 6 5 3 1 1
## [5005] 5 1 5 2 6 6 6 3 6 1 2 2 6 4 6 1 4 5 5 1 3 5 5 2 4 1 1 3 3 2 2 3 5 3 4 5
## [5041] 6 5 4 4 1 3 1 4 2 5 1 2 6 6 1 5 2 1 5 1 4 5 4 3 4 6 2 2 4 5 3 2 3 4 3 6
## [5077] 2 2 3 1 6 3 2 5 6 5 5 1 6 2 2 6 2 1 5 4 3 2 5 5 5 2 2 4 3 4 1 1 5 2 2 5
## [5113] 5 3 2 4 4 5 3 4 5 2 2 4 3 4 3 6 6 5 2 3 4 3 6 5 5 1 4 2 2 5 3 3 5 4 3 2
## [5149] 3 4 6 1 1 6 1 5 3 1 1 1 5 5 4 4 4 2 4 4 6 6 1 6 1 3 2 4 1 5 3 4 4 5 1 5
## [5185] 5 2 1 6 5 6 3 5 5 6 3 5 6 4 3 1 6 6 3 6 2 6 2 5 6 3 4 5 5 3 3 3 3 4 4 1
## [5221] 2 1 3 5 3 2 5 1 4 1 2 1 5 6 3 1 5 5 1 5 5 6 6 1 4 4 2 6 2 1 5 2 1 5 3 5
## [5257] 2 4 5 3 4 3 4 3 2 6 4 3 3 1 4 4 2 2 1 5 3 2 6 1 3 3 5 1 2 1 1 4 6 2 2 4
## [5293] 6 6 4 4 6 1 5 2 4 4 6 2 6 1 4 6 5 4 6 4 4 4 2 4 3 4 5 5 2 4 5 4 2 1 3 2
## [5329] 4 2 5 3 6 6 5 3 6 1 2 3 3 4 2 4 6 6 6 4 5 4 3 3 5 6 6 6 2 6 3 2 2 4 6 5
## [5365] 6 5 2 6 2 4 2 3 4 1 5 1 6 2 1 2 2 2 5 1 2 3 2 1 5 1 5 2 4 4 4 6 2 5 1 6
## [5401] 5 2 5 2 6 2 3 6 2 6 4 5 5 6 1 2 5 3 2 3 2 5 4 1 3 2 3 2 4 2 3 2 2 3 5 2
## [5437] 5 3 6 6 6 6 2 6 5 5 4 4 5 3 5 6 1 3 5 5 3 6 5 6 2 2 6 3 1 4 2 3 6 6 5 6
## [5473] 2 4 1 1 3 1 1 4 5 6 3 3 4 3 4 3 4 3 1 2 3 2 2 1 6 5 5 5 2 2 1 1 1 2 5 6
## [5509] 1 3 4 5 2 3 2 3 6 6 3 4 2 1 1 5 6 2 6 6 5 1 2 6 4 6 4 2 5 2 6 5 1 1 3 2
## [5545] 3 3 5 2 6 4 5 6 2 2 2 5 6 1 5 6 3 3 2 4 1 3 4 2 3 5 2 2 2 4 3 4 4 1 3 6
## [5581] 5 4 6 5 2 3 2 5 3 4 4 6 5 4 1 4 4 3 3 6 4 1 4 4 3 6 5 2 2 6 6 2 6 5 5 2
## [5617] 2 2 4 5 6 3 1 2 3 2 3 1 3 2 2 1 6 2 1 6 3 1 6 3 6 4 2 6 2 5 4 6 6 3 3 2
## [5653] 1 6 3 3 1 2 4 1 1 5 6 1 1 5 1 5 1 5 1 6 6 1 2 1 1 3 6 3 1 3 1 6 2 4 2 1
## [5689] 2 4 1 4 3 3 1 5 2 6 3 5 3 4 4 1 1 1 2 6 6 5 3 3 5 4 2 5 6 2 1 6 4 6 3 2
## [5725] 6 4 4 1 1 3 6 4 1 3 2 1 4 4 1 3 3 2 6 5 1 4 3 3 3 5 4 3 4 4 3 6 6 4 1 2
## [5761] 3 1 4 3 6 3 4 3 4 2 6 3 6 1 6 5 3 4 4 3 3 5 3 6 1 6 1 2 4 4 1 1 2 1 2 5
## [5797] 3 1 1 3 1 3 3 4 6 4 5 2 1 5 6 3 5 3 6 2 6 5 2 1 6 6 5 6 5 1 3 5 3 6 5 6
## [5833] 5 2 6 4 2 2 3 5 6 1 6 3 1 5 6 4 4 2 5 6 4 2 5 5 3 2 2 5 1 3 6 3 5 2 5 5
## [5869] 5 4 4 3 2 3 1 3 6 5 4 2 4 4 1 6 5 3 5 4 5 1 3 2 1 2 3 3 1 6 2 6 6 2 6 3
## [5905] 5 2 4 6 5 5 5 6 6 6 5 5 5 5 1 5 2 2 2 5 6 4 5 6 5 6 3 5 5 3 3 1 3 3 5 6
## [5941] 4 5 2 6 4 3 3 4 5 4 2 2 5 4 2 3 2 4 3 4 5 4 1 1 1 4 3 4 6 2 2 2 6 5 6 2
## [5977] 3 2 6 3 6 4 1 4 2 6 1 1 3 5 2 1 4 2 5 1 6 5 1 2 1 5 5 3 6 6 3 4 1 5 6 1
## [6013] 4 3 3 5 3 4 5 6 5 4 3 5 1 5 5 1 2 2 4 3 6 2 3 3 2 1 2 2 3 6 6 4 1 2 1 3
## [6049] 3 5 4 3 1 2 4 5 2 4 1 6 5 4 1 4 5 2 6 2 5 6 5 6 3 2 5 1 6 1 2 5 5 3 3 6
## [6085] 3 6 6 3 5 3 3 5 6 3 3 4 1 6 5 4 3 1 5 5 3 4 5 5 1 2 6 6 6 3 3 3 6 4 1 2
## [6121] 4 5 1 4 4 4 6 5 1 3 4 4 3 1 1 3 6 3 5 3 5 3 2 6 2 5 5 5 2 5 5 3 3 3 6 2
## [6157] 3 4 3 4 3 2 5 1 1 6 2 2 1 1 1 6 6 2 2 3 1 1 2 6 1 1 5 5 3 5 6 5 4 5 1 3
## [6193] 2 3 6 5 1 1 5 3 2 5 1 6 6 6 3 5 1 6 1 5 2 3 3 5 6 2 3 3 2 5 6 2 6 1 4 3
## [6229] 1 5 6 3 4 1 5 3 2 5 6 2 6 2 6 4 2 4 1 4 4 6 1 5 5 3 6 5 3 2 1 1 2 6 6 1
## [6265] 1 2 3 6 6 2 5 4 1 2 3 2 2 2 3 5 6 5 5 4 4 1 3 5 6 1 1 4 6 2 4 1 4 6 3 4
## [6301] 2 1 3 4 1 6 4 4 2 3 3 2 3 6 6 1 3 3 6 4 1 5 1 6 6 2 3 6 4 2 4 2 6 2 2 3
## [6337] 3 1 4 1 1 5 6 4 4 2 5 1 3 6 2 3 2 6 4 1 3 1 1 2 5 4 5 2 3 1 5 4 2 6 4 6
## [6373] 5 5 2 2 1 5 5 5 1 3 6 4 3 4 5 1 2 3 6 3 5 2 5 5 5 2 6 4 2 1 2 6 5 4 4 2
## [6409] 3 3 4 5 1 1 5 2 3 6 4 2 4 5 6 3 2 6 4 2 1 1 2 4 6 3 1 4 3 4 5 2 1 3 5 1
## [6445] 3 4 2 5 4 4 5 6 3 5 4 1 1 1 1 6 4 3 1 5 5 6 1 6 2 5 6 4 3 2 2 2 2 4 3 3
## [6481] 1 4 2 6 1 3 2 6 6 6 5 5 4 6 4 1 4 5 2 2 6 6 4 5 3 3 5 1 2 3 3 5 1 3 4 3
## [6517] 5 4 3 1 6 5 6 2 6 4 1 2 1 6 3 2 5 2 2 4 2 1 4 6 2 6 6 1 6 3 2 6 6 1 4 5
## [6553] 4 5 5 5 6 4 2 4 2 1 6 3 4 6 3 2 5 3 1 6 6 5 2 2 6 5 1 6 1 6 6 3 3 6 6 2
## [6589] 3 4 3 6 5 2 5 5 4 3 3 1 6 6 6 2 2 1 6 1 3 6 1 4 5 5 6 6 3 1 1 2 3 6 1 1
## [6625] 1 6 2 2 1 5 1 3 2 2 5 5 5 5 4 3 2 2 5 5 1 2 5 5 3 3 3 1 2 2 5 3 2 1 3 6
## [6661] 6 2 6 5 5 3 2 2 3 5 1 1 6 1 5 6 3 2 1 3 1 3 5 2 4 3 6 6 6 2 3 3 4 1 4 6
## [6697] 5 5 3 6 6 5 5 2 4 2 5 3 1 5 6 3 5 1 2 5 3 2 5 2 4 6 4 3 1 2 4 1 4 5 1 6
## [6733] 1 6 4 2 2 1 6 3 5 6 6 1 4 3 2 5 5 1 2 4 4 4 2 6 5 2 3 1 1 1 2 3 4 6 2 3
## [6769] 3 5 5 3 5 4 3 6 5 3 6 6 1 1 5 4 6 5 6 4 2 6 4 2 4 1 5 3 5 4 6 4 5 1 3 5
## [6805] 5 2 4 1 2 6 1 5 1 6 6 3 4 3 5 5 2 5 5 5 1 1 5 4 3 4 1 5 6 5 1 3 1 4 2 4
## [6841] 2 4 4 4 5 1 1 5 5 2 6 3 1 5 5 1 6 3 5 6 2 2 5 5 2 5 5 5 3 6 6 5 3 1 4 5
## [6877] 2 1 2 2 2 3 4 1 1 3 2 4 6 2 3 5 6 6 1 5 4 1 5 2 5 6 5 4 1 2 6 1 5 2 3 6
## [6913] 3 6 5 1 5 2 6 5 2 1 2 6 1 5 4 3 4 4 4 3 5 2 5 3 2 3 1 2 6 5 4 5 5 3 2 1
## [6949] 2 2 5 6 2 5 4 6 4 1 2 3 6 4 2 2 2 3 1 5 1 6 1 6 2 5 1 5 5 4 3 3 3 5 4 5
## [6985] 1 5 4 5 4 6 6 3 4 4 5 1 1 6 6 6 1 6 2 3 2 4 6 1 4 2 6 1 2 3 3 3 4 6 3 5
## [7021] 2 6 4 2 5 4 5 3 3 6 4 6 3 4 5 3 3 1 2 3 6 6 6 5 4 5 5 2 3 3 1 4 3 6 2 6
## [7057] 1 3 4 5 6 1 5 2 3 4 2 3 5 3 6 1 4 1 4 3 6 3 3 5 6 6 3 5 1 6 1 1 1 1 1 1
## [7093] 4 1 3 1 3 2 4 3 4 2 6 5 4 6 3 5 1 6 5 3 1 4 1 5 2 6 4 3 5 2 6 2 1 6 5 1
## [7129] 3 3 3 6 3 4 5 2 3 6 4 5 3 3 4 2 1 6 6 2 6 5 5 5 1 5 6 6 4 1 4 6 4 1 5 4
## [7165] 2 5 3 2 6 2 2 6 2 3 4 5 1 5 4 4 2 4 6 2 3 1 1 5 4 4 1 1 2 5 6 3 5 1 3 2
## [7201] 2 3 3 6 1 3 1 6 3 5 4 5 3 3 2 3 6 2 6 6 4 2 1 3 6 4 4 3 4 4 5 3 6 1 6 1
## [7237] 4 1 1 6 3 3 5 1 1 5 6 5 2 3 3 1 3 1 1 4 5 3 6 5 1 5 4 6 2 3 4 3 2 6 1 1
## [7273] 1 4 4 4 3 3 5 6 5 1 4 1 6 6 1 1 2 3 6 1 1 3 1 5 5 1 1 4 4 1 5 4 1 5 5 5
## [7309] 4 1 6 5 5 6 3 3 1 3 5 2 6 4 6 6 5 6 3 6 6 4 1 4 2 6 6 2 6 3 5 3 4 2 6 4
## [7345] 2 2 2 2 5 3 1 3 5 5 1 3 3 4 6 2 2 6 5 3 6 1 3 6 3 1 3 6 1 5 4 1 1 1 2 6
## [7381] 1 3 3 4 2 2 6 5 2 2 4 6 3 4 3 4 6 6 6 4 6 3 1 4 5 2 6 5 6 5 2 1 1 3 4 6
## [7417] 4 3 3 3 5 3 1 1 4 2 3 2 6 1 6 1 5 3 2 5 6 6 2 5 5 4 4 6 1 2 4 2 3 3 3 3
## [7453] 2 1 2 4 6 2 4 6 6 1 5 1 3 4 6 3 5 3 4 2 1 4 2 5 3 3 5 1 6 2 1 2 5 3 3 4
## [7489] 5 6 6 5 5 3 3 4 3 5 2 1 2 2 2 5 5 3 4 1 4 3 6 1 1 1 6 1 1 1 1 2 1 2 4 2
## [7525] 4 6 3 3 2 3 2 1 5 3 4 3 2 1 4 2 3 2 5 6 3 4 6 3 3 1 6 1 5 5 6 4 1 6 3 6
## [7561] 3 2 5 1 4 4 2 3 2 1 2 1 3 2 1 5 3 6 3 4 3 3 2 1 1 1 4 6 6 6 5 3 1 2 3 4
## [7597] 6 1 3 5 5 4 2 6 5 2 6 4 1 4 5 5 1 3 2 3 1 1 3 5 1 2 6 5 2 1 4 4 2 4 5 6
## [7633] 6 6 3 1 2 1 3 5 3 4 6 6 2 5 3 1 6 2 5 6 6 2 6 3 3 5 5 5 2 3 1 5 3 1 4 1
## [7669] 2 3 6 1 3 6 2 6 6 6 1 6 4 1 5 4 2 6 5 6 1 6 1 1 3 6 4 3 4 4 2 4 1 1 1 6
## [7705] 5 5 2 5 6 1 6 2 3 2 2 1 5 6 6 4 6 2 1 1 1 4 5 6 5 3 6 3 4 6 1 3 6 2 3 1
## [7741] 6 1 5 6 3 2 5 2 6 1 1 5 3 6 6 6 3 3 5 1 2 3 1 5 4 5 4 4 4 5 4 1 3 2 6 3
## [7777] 1 3 1 4 4 2 5 3 1 6 1 3 1 1 4 6 5 2 5 1 4 6 6 6 5 6 1 4 2 3 1 3 4 5 2 3
## [7813] 1 2 6 3 1 2 4 2 2 2 3 2 2 3 3 4 2 6 1 3 3 5 6 4 4 6 3 4 1 5 2 2 4 4 1 4
## [7849] 3 3 5 2 6 5 5 4 4 6 5 1 6 6 5 6 1 2 4 2 3 5 5 2 4 2 5 1 2 4 5 4 2 5 6 5
## [7885] 5 5 5 4 1 3 3 4 5 2 2 3 1 5 1 3 6 1 4 3 1 1 2 2 4 6 4 2 3 1 3 4 1 4 6 1
## [7921] 1 3 5 5 1 6 3 5 5 5 6 1 6 6 2 3 6 1 4 4 4 4 3 4 4 1 5 5 2 3 4 3 3 6 2 5
## [7957] 6 6 4 1 1 6 3 5 5 5 2 5 4 2 4 6 4 2 1 4 5 4 5 2 1 1 6 5 1 3 6 3 2 1 6 3
## [7993] 4 4 3 1 6 2 1 2 6 1 1 2 6 5 5 1 6 2 2 1 2 6 2 1 5 3 1 2 3 3 5 2 6 5 1 1
## [8029] 2 3 4 6 1 3 2 2 4 5 2 2 3 1 4 3 6 2 6 2 2 3 5 3 6 4 5 6 6 3 3 6 2 1 4 3
## [8065] 1 3 2 1 6 5 1 2 5 2 6 1 6 3 4 1 1 1 2 3 2 6 1 6 1 3 4 5 1 6 5 2 4 2 2 6
## [8101] 4 4 3 5 5 6 6 6 3 6 4 2 4 3 1 1 6 2 5 2 1 1 6 5 4 2 1 1 3 2 4 5 6 3 2 1
## [8137] 3 6 1 2 3 1 1 5 4 1 6 6 2 5 6 4 6 2 3 2 1 5 1 5 4 2 3 4 1 5 5 1 1 2 6 4
## [8173] 6 4 2 6 6 2 4 3 1 6 3 1 1 4 1 5 6 4 5 3 3 1 5 1 2 6 2 2 4 4 5 6 4 1 4 1
## [8209] 4 4 5 5 2 3 3 2 5 2 5 4 6 6 4 2 3 1 4 6 5 4 4 2 5 5 6 2 4 5 5 2 4 6 6 5
## [8245] 4 2 4 1 5 3 2 4 1 5 3 6 3 6 5 3 2 1 6 4 3 4 6 3 2 6 1 6 2 6 1 4 6 4 2 4
## [8281] 5 3 5 5 4 5 2 1 3 4 6 3 5 5 5 2 1 4 6 6 5 3 2 1 1 3 3 3 2 5 3 2 6 1 2 3
## [8317] 3 6 3 5 4 2 3 5 2 5 5 6 4 6 5 1 1 4 6 2 2 3 1 5 5 5 6 6 3 5 5 6 1 6 6 1
## [8353] 2 5 2 5 6 4 3 3 3 2 6 5 1 3 2 1 1 5 1 5 5 6 2 3 2 5 4 6 5 1 1 1 4 6 5 1
## [8389] 1 4 1 6 3 4 1 1 1 1 4 6 6 2 4 1 5 3 2 3 2 2 3 1 6 1 4 4 5 5 5 2 5 6 5 1
## [8425] 4 5 5 1 4 1 2 5 6 3 5 1 5 6 2 6 5 1 4 3 6 1 5 2 4 4 5 1 4 3 5 2 3 1 4 6
## [8461] 4 6 6 1 1 5 1 1 5 1 2 2 2 2 5 6 1 2 6 4 2 2 6 3 2 5 5 3 3 3 3 2 2 3 5 5
## [8497] 3 5 6 4 4 3 1 3 2 5 2 3 6 5 5 1 2 5 1 3 4 1 1 4 1 2 4 1 5 4 3 6 3 5 2 2
## [8533] 4 5 4 3 5 6 1 5 3 3 1 5 3 4 1 1 3 2 4 4 1 4 2 5 3 3 4 5 6 4 1 3 1 6 3 6
## [8569] 1 5 1 4 6 6 3 2 4 5 1 4 2 3 1 5 2 3 4 6 2 6 4 4 2 4 4 1 2 2 3 6 2 1 3 4
## [8605] 3 5 5 5 3 6 5 5 2 1 4 2 6 1 3 1 4 1 4 6 1 3 5 6 6 3 3 2 5 1 6 3 2 4 6 4
## [8641] 4 3 5 6 3 1 3 4 3 3 6 3 4 1 5 6 5 5 4 6 3 1 6 6 1 2 6 1 5 6 2 5 4 3 1 3
## [8677] 6 5 4 5 3 1 2 2 4 2 1 5 5 3 6 1 2 1 1 1 6 4 6 2 3 4 1 3 3 5 1 6 1 6 5 2
## [8713] 6 2 3 4 4 4 3 6 1 2 6 2 3 6 1 4 6 3 3 6 6 1 1 3 2 4 6 6 3 3 5 1 1 2 5 1
## [8749] 1 3 4 4 2 3 3 1 4 1 2 3 5 3 4 4 3 1 6 5 3 3 5 5 6 5 6 1 4 3 2 4 6 2 3 2
## [8785] 2 3 2 6 1 6 3 2 2 5 4 1 6 5 3 5 5 5 5 4 5 4 3 4 1 2 2 1 5 5 2 1 3 5 4 5
## [8821] 3 3 1 6 1 1 2 4 4 5 1 5 6 4 3 5 1 1 3 1 1 2 6 6 3 3 3 1 3 5 2 5 4 2 2 1
## [8857] 6 1 6 5 3 1 3 1 6 6 3 1 6 2 2 4 2 4 2 1 2 1 6 5 1 2 5 2 4 2 3 2 2 2 5 2
## [8893] 4 2 1 4 6 6 4 6 1 2 3 6 3 2 3 1 2 5 2 6 4 2 5 4 4 2 1 6 6 5 5 5 6 1 5 6
## [8929] 4 6 5 2 2 1 1 3 2 3 4 3 2 5 2 1 3 2 5 1 5 2 4 5 4 6 5 3 1 4 5 5 6 2 1 1
## [8965] 6 3 3 3 5 2 4 3 3 2 6 6 2 5 3 2 1 5 5 3 5 5 6 6 2 1 2 4 6 1 4 2 4 5 4 3
## [9001] 6 6 2 4 5 5 2 2 1 2 2 6 2 4 3 5 4 4 6 3 4 4 3 1 6 2 1 1 5 4 6 2 5 4 2 2
## [9037] 4 4 6 1 4 2 5 6 3 1 6 5 6 2 5 2 6 1 3 5 6 2 6 1 2 3 4 5 6 6 2 6 3 3 6 3
## [9073] 1 3 5 3 3 2 4 5 5 2 2 3 5 1 2 2 3 1 4 2 6 5 6 2 4 3 4 1 6 3 1 5 6 1 2 3
## [9109] 1 1 5 6 1 4 2 3 6 2 2 4 3 2 3 2 3 2 4 5 3 5 2 1 5 6 4 3 1 6 2 4 5 2 3 3
## [9145] 1 2 5 5 5 3 3 4 6 2 2 5 5 1 4 3 3 5 4 5 5 5 6 4 1 3 2 4 5 2 5 3 4 6 6 3
## [9181] 1 2 6 2 1 5 2 1 2 1 5 2 1 3 2 5 3 2 3 2 6 4 1 2 4 2 1 5 2 5 6 6 3 5 3 3
## [9217] 1 2 1 4 6 5 2 1 5 4 6 1 4 5 2 1 2 4 2 1 2 4 1 6 2 6 5 4 1 2 5 2 3 3 3 2
## [9253] 4 4 2 2 1 5 6 3 5 5 6 1 3 2 6 1 5 2 2 1 1 3 3 4 5 6 5 3 3 3 3 1 4 2 1 4
## [9289] 5 6 5 6 6 5 3 5 6 2 5 6 5 2 2 3 6 2 5 1 5 6 1 1 3 4 2 2 4 1 5 4 3 6 1 5
## [9325] 6 6 1 1 2 3 1 6 6 4 1 2 4 1 4 6 2 6 1 4 5 1 5 3 4 2 6 4 4 4 3 5 2 4 4 4
## [9361] 2 5 6 5 6 2 5 6 6 6 3 3 1 6 5 6 5 1 6 2 2 3 1 4 2 6 1 3 4 4 5 6 3 4 5 2
## [9397] 5 5 4 2 4 4 6 1 5 1 4 4 2 6 2 1 5 2 6 3 3 6 5 2 2 5 2 5 3 6 1 1 1 2 2 3
## [9433] 6 1 2 6 3 4 2 6 6 3 4 1 6 2 4 3 6 5 2 6 3 6 2 6 2 2 1 6 5 2 1 5 3 5 6 6
## [9469] 5 5 5 3 4 6 3 5 6 5 3 5 1 1 5 1 6 2 6 5 4 2 3 4 4 5 5 3 1 3 5 4 6 5 4 5
## [9505] 4 1 2 3 2 2 4 3 5 1 1 4 5 2 6 3 4 3 1 2 6 5 5 5 4 5 3 5 5 4 3 4 2 3 4 6
## [9541] 4 3 5 5 1 6 3 4 2 2 2 1 4 6 4 4 2 3 3 3 5 5 3 6 5 3 6 5 4 3 4 1 3 1 5 4
## [9577] 2 4 2 6 6 5 1 6 6 4 3 1 2 1 5 5 3 1 2 3 1 1 1 6 4 2 6 3 2 6 3 2 4 3 4 2
## [9613] 5 2 2 2 1 3 5 5 5 5 6 2 5 2 3 2 2 1 1 6 3 3 2 1 4 3 6 2 5 5 1 1 5 2 2 2
## [9649] 2 6 3 1 4 4 2 3 6 1 5 6 1 6 5 4 1 1 6 3 4 5 5 5 6 4 5 2 2 3 4 5 6 2 2 4
## [9685] 3 1 2 4 6 5 5 5 6 5 4 6 5 5 4 5 6 3 6 5 4 2 1 5 5 6 5 1 3 3 2 2 1 1 6 2
## [9721] 6 6 3 3 3 1 6 5 6 6 2 4 5 1 2 6 1 4 3 3 6 5 2 5 4 4 2 5 5 4 5 2 4 5 1 2
## [9757] 5 6 2 1 5 1 2 5 5 1 3 2 5 3 1 5 2 3 2 6 1 4 3 3 6 1 5 4 4 6 5 3 2 2 5 6
## [9793] 5 4 4 4 3 1 6 6 5 3 3 6 3 5 2 2 5 2 5 3 1 5 6 4 2 1 1 6 5 5 6 4 4 1 6 3
## [9829] 5 2 3 5 2 4 3 1 3 1 4 3 3 1 3 6 2 6 6 2 5 2 4 5 2 6 4 4 3 3 5 4 3 6 6 4
## [9865] 1 2 2 3 1 4 5 4 4 2 1 4 3 5 5 2 1 1 4 6 1 4 2 3 1 6 5 2 2 3 1 5 5 3 3 2
## [9901] 6 5 3 2 1 2 5 5 4 4 3 3 4 2 5 4 6 4 1 3 2 6 3 3 1 1 2 2 1 2 1 5 6 6 5 1
## [9937] 6 6 5 4 5 6 2 4 2 5 3 5 6 6 6 6 2 2 2 1 1 4 2 1 3 4 3 6 5 1 2 1 2 1 5 3
## [9973] 2 4 2 1 2 6 5 6 5 1 3 1 1 1 3 3 4 1 5 6 5 1 1 1 2 4 6 5Q) How frequently did the die land on a “six”?
- 0.1616
FYI you can find this probability with the R code:
sum(obbservations == 6)/trialsExample: Flipping an unfair coin
Q) How could we figure out the probability that our unfair coin lands on a “heads”?
Note: There is another way to use the laws of probability. For example, perhaps I’m a magician and I’m flipping a coin and you don’t know whether or not the coin is fair. You could express your doubts with a subjective probability.
Subjective probability - uses the laws of probability to describe a personal degree of (un)certainty, unrelated to any kind of long-run behavior.
3. Laws of Probability
1) A probability is a number between 0 and 1.
\[0 \leq Pr(A) \leq 1, \quad\text{for $A \in S$}\]
2) The probability of the set of all possible outcomes of a trial is 1.
\[Pr(S)=1\]
3) The probability of an event not occuring is equal to 1 minus the probability the event does occur.
\[Pr(A^{C}) = 1 - Pr(A)\]
4) If an event \(A\) is disjoint from another event \(B\), then the probability that one event or the other event occurs is the sum of their two individual probabilities.
\[Pr(A\text{ or }B) = Pr(A) + Pr(B), \quad\text{when }Pr(A\text{ and }B) = 0\]
5) If an event \(A\) is independent of another event \(B\), then the probability that both events occur is the product of the probabilities of the two individual events.
\[Pr(A\text{ and } B) = Pr(A)\times Pr(B), \quad\text{when $A$ and $B$ are independent events}\]
Example: Selecting colored balls from a box
Two boxes contain red and blue balls. Box 1 has 14 blue balls and 4 red balls. Box 2 has 10 blue balls and 2 red balls.
The random phenomena we are considering is drawing a single ball from each of the two boxes without looking.
What is the sample space for this random phenomena?
Are you more likely to choose a blue ball from Box 1 or Box 2?
What is the probability that you select a blue ball from Box 1 and a blue ball from Box 2